Download Proof Theory: From Arithmetic to Set Theory

Proof Theory: From Arithmetic to Set Theory
Michael Rathjen
Accompanying notes for a course given at the Nordic Spring School,
Nordfjordeid, 27–30 May 2013
Contents
• A brief history of proof theory
• Sequent calculi for classical and intuitionistic logic, Gentzen’s Hauptsatz: Cut
elimination
• Consequences of the Hauptsatz: Subformula property, Herbrand’s Theorem,
existence and disjunction property, geometric theories
• Ordinal functions and representations up to Γ0
• Ordinal analysis of Peano arithmetic, PA, and some subsystems of second
order arithmetic.
• Limits for the deducibility of transfinite induction
• Kripke-Platek set theory, KP.
• The Bachmann-Howard ordinal
• KP goes infinite, RS.
• Impredicative cut elimination theorem
• Interpreting KP in RS
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1
A short and biased history of logic till 1938
• Logical principles - principles connecting the syntactic structure of sentences
with their truth and falsity, their meaning, or the validity of arguments in
which they figure - can be found in scattered locations in the work of Plato
(428–348 B.C.).
• The Stoic school of logic was founded some 300 years B.C. by Zeno of Citium
(not to be confused with Zeno of Elea). After Zeno’s death in 264 B.C., the
school was led by Cleanthes, who was followed by Chrysippus. It was
largely through the copious writings of Chrysippus that the Stoic school became established, though many of these writings have been lost.
• The patterns of reasoning described by Stoic logic are the patterns of interconnection between propositions that are completely independent of
what those propositions say.
• The first known systematic study of logic which involved quantifiers, components such as “for all” and “some”, was carried out by Aristotle (384–322
B.C.) whose work was assembled by his students after his death as a treatise
called the Organon, the first systematic treatise on logic.
• Aristotle tried to analyze logical thinking in terms of simple inference rules
called syllogisms. These are rules for deducing one assertion from exactly
two others.
• An example of a syllogism is:
P 1. All men are mortal.
P 2. Socrates is a man.
C. Socrates is mortal.
• In the case of the above syllogism, it is obvious that there is a general pattern,
namely:
P 1. All M are P .
P 2. S is a M .
C. S is P .
• Some of the other syllogisms Aristotle formulated are less obvious. E.g.
P 1. No M is P .
P 2. Some S is M .
C. Some S is not P .
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• Aristotle appears to have believed that any logical argument can, in principle,
be broken down into a series of applications of a small number of syllogisms.
He listed a total of 19.
• The syllogism was found to be too restrictive (much later).
• For almost 2000 years Aristotle was revered as the ultimate authority on logical
matters.
Bachelors and Masters of arts who do not follow Aristotle’s philosophy are subject to a fine of five shillings for each point of divergence,
as well as for infractions of the rules of the ORGANON.
– Statuses of the University of Oxford, fourteenth century.
When did Modern Logic start?
• Aristotle’s logic was very weak by modern standards.
• The ideas of creating an artificial formal language patterned on mathematical notation in order to clarify logical relationships - called characteristica
universalis - and of reducing logical inference to a mechanical reasoning process in a purely formal language - called calculus rationatur - were due to
Gottfried Wilhelm Leibniz (1646-1716).
• Leibniz’s contributions include arithmetization of syllogistic, a theory of relations, modal logic and logical grammar.
• Much of it published posthumously 1903 by Couturat Opuscules et fragment
inédit de Leibniz.
• Logic as we know it today has only emerged over the past 140 years.
• Chiefly associated with this emergence is Gottlob Frege (1848–1925). In his
Begriffsschrift 1879 (Concept Script) he invented the first programming
language.
• His Begriffsschrift marked a turning point in the history of logic. It broke new
ground, including a rigorous treatment of quantifiers and the ideas of functions
and variables.
• Frege wanted to show that mathematics grew out of logic.
• Charles Peirce (1839–1914) is another pioneer of modern logic.
• Another strand is Algebraic logic which stresses logic as a calculus: Augustus De Morgan (1806–1871), George Boole (1815–1864), Ernst Schröder
(1841–1902).
• Modern logic was codified in Principia Mathematica (1910,1912,1913) by
Bertrand Russell (1872–1970) and Alfred N. Whitehead (1861–1947).
3
The Origins of Proof Theory (Beweistheorie)
• David Hilbert (1862–1943)
• Hilbert’s second problem (1900): Consistency of Analysis
• Hilbert’s Programme (1922,1925)
The Grundlagenkrise: the usual suspects
• Inconsistency in Frege’s Grundlagen.
• Cantor had already observed that in set theory the unrestricted Comprehension Principle (CP) leads to contradictions. CP allows one to build
sets by collecting all the sets having in common a property P to form a new
set
{x | P (x)}.
• Russell’s Paradox (1901)
• Hermann Weyl: “Über die neue Grundlagenkrise in der Mathematik” (1921)
19th century: Growth of the subject
• Beginning 19th century: mathematics was concrete, constructive, algorithmic
• End of 19th century: Much abstract, non-constructive, non-algorithmic mathematics was under development
growing preference for short conceptual non-computational proofs over long
computational proofs.
• Non-euclidian geometries: statements can be true in one geometry and
false in another.
• But also consolidation: (More) rigorous foundations of analysis: Cauchy (17891857), Bolzano (1781-1848), Weierstrass (1815-1897)
4
New (non-constructive) proof methods
• Abstract notion of function (In Euler’s time functions were explicitly defined via an analytic expression)
• Indirect existence proofs (Hilbert’s Basis Theorem)
• Zermelo’s proof that R (the reals) can be well-ordered (1904)
Axiom of Choice
Let I be a set. Suppose that Ai is a non-empty set for each i ∈ I. Then
there exists a function
[
f : I −→
Ai
i∈I
such that
f (i) ∈ Ai
holds for all i ∈ I.
Borel, Baire, Lebesgues against the Axiom of Choice 1905
Borel: It seems to me that the objection against it is also valid for
every reasoning where one assumes an arbitrary choice made an uncountable number of times, for such reasoning does not belong
in mathematics.
Acceptance of AC
• By the 1930s AC was widely accepted.
• With AC, every vector space has a basis.
• Let V, W be a vector spaces over same field, u ∈ V, w ∈ W and u, w 6= 0.
Then there is a linear mapping f : V → W such that f (v) = w.
Reactions and Cures
• Brouwer (1908) rejects the law of excluded middle
(A ∨ ¬A for arbitrary statements A)
Intuitionistic Mathematics
• Russell (1908) Vicious Circle Principle
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• H. Weyl (1885-1955) criticizes impredicative set formation principles
Mathematics .. house build on sand (1918)
Hilbert’s way out
• Platonists, Logicists and Intuitionists seem to agree that a mathematical concept, or sentence, or a theory is acceptable (or properly understood) only if all terms which occur in it can be interpreted directly.
• By contrast, the formalist holds that direct interpretability is not a necessary condition for the acceptability of a mathematical theory.
To understand a theory means to be able to follow its logical development
and not, necessarily, to interpret, or give a denotation for, its individual
terms.
Hilbert’s two-tiered approach
1. Interpreted (material ”inhaltlich”) Mathematics: Basic rules of reasoning
and arithmetic whose validity is self-evident.
2. Uninterpreted (or formal) mathematics obtained by the adjunction of
“ideal” (uninterpreted) elements to material ”inhaltliche” mathematics
In Hilbert’s case, interpreted mathematics was finitistic mathematics wherein
reference to actual infinite sets was tabu.
Hilbert’s Program (1922,1925)
• I. Codify the whole of mathematical reasoning in a formal theory T.
• II. Prove the consistency of T by finitistic means.
• “No one shall drive us from the paradise which Cantor has created for us.”
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Finitism
• The exact meaning of “finitistic means” was never precisely delineated by
Hilbert.
• Finitistic means form the basis of any scientific reasoning.
• They do not refer to the actual infinite and do not include any objectionable
proof methods.
Hilbert’s Ontology
Real Objects:
Ideal objects:
the natural numbers,
finite strings of symbols
(something a computer can deal with)
the other mathematical objects:
abstract functions, choice functions, Hilbert spaces, ultrafilters, etc.
• Real objects are the main concern of mathematicians.
They exist.
• Ideal/abstract objects exist merely as a façon de parler. But they are
important for the progress of mathematics.
The method of ideal elements
• Solve a mathematical problem regarding a specific mathematical structure by
adding new ideal elements to the structure.
• Hilbert: The method of ideal elements is of great importance to the progress
of mathematical research.
Examples
Elementary Geometry → Points and lines at ∞
→ Projective Geometry
Elementary number theory → number fields, ideals
→ algebraic number theory
Analysis/number theory → Ultrafilter
→ Set theory
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Indispensable condition
• Hilbert: Es gibt nämlich eine Bedingung, eine einzige, aber auch absolut
notwendige, an die die Anwendung der Methode der idealen Elemente geknüpft
ist, und diese ist der Nachweis der Widerspruchsfreiheit: die Erweiterung
durch Zufügung von Idealen ist nämlich nur dann statthaft, wenn also die
Beziehungen, die sich bei Elimination der idealen Gebilde für die alten Gebilde
herausstellen, stets im alten Bereiche gültig sind.
• There is just one condition, albeit an absolutely necessary one, connected with
the method of ideal elements. That condition is a proof of consistency, for
the extension of a domain by the addition of ideal elements is legitimate only
if the extension does not cause contradictions to appear in the old, narrower
domain, or, in other words, only if the relations that obtain among the old
structures when the ideal structures are deleted are always valid in the old
domain.
• Another reading of Hilberts Programme:
Elimination of ideal elements.
Maybe we should refrain from ontological talk
• Abraham Robinson (1918-74):
Non-standard analysis (1966)
•
this book ... appears to affirm the existence of all sorts of infinitary
entities.
However, from a formalist point of view we may look at our theory syntactically and may consider that what we have done is to
introduce new deductive procedures rather than new mathematical entities.
Mathematical statements
Real statements
A
A
A
A
Ideal statements
Real statements are of the following forms:
∀x1 · · · ∀xr f (x1 , .., xr ) = g(x1 , .., xr );
∀x1 · · · ∀xr f (x1 , .., xr ) 6= g(x1 , .., xr );
∀x1 · · · ∀xr f (x1 , .., xr ) ≤ g(x1 , .., xr )
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where f, g are basic functions (polynomials) on the naturals.
Examples of real statements
• Goldbach’s conjecture: Every even number n > 2 is the sum of two primes.
(Confirmed up to at least 1018 ).
• Vinogradov’s Three Primes Theorem 1937: Every odd integer > 1013000
is the sum of three primes.
• Fermat’s conjecture ( Wiles’ Theorem 1995) :
“For all naturals a, b, c, n, if a · b · c 6= 0 and n > 2 then
an + bn 6= cn .
• Riemann hypothesis All non-trivial zeros s of ζ satisfy Re(s) = 12 .
• Four colour theorem
Ideal statements
• The axiom of choice.
• Every vector space has a base.
• If R is a noetherian ring, then so is the polynomial ring R[X].
• (Schröder-Berstein Theorem) If f : X → Y and g : Y → X are both
injective functions, then there exists a 1-1 correspondence between X and Y .
Example of a real statement proved by using ideal elements
Theorem: 1.1 (Hadamard, de La Vallée Poussin 1896) Prime number
theorem
π(x)
lim x = 1
x→∞
ln(x)
where π(x) = number of prime numbers ≤ x.
The original proof used contour integration of curves over C.
Atle Selberg and Paul Erdös (1949) found proofs using only the means of
elementary number theory.
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Hilbert’s Conservation Programme
• A consequence of Hilbert’s Programme
• Hilbert’s hope:
If a real statement Ψ is provable in non-finitistic mathematics, then Ψ can also
be proved by purely finitistic means.
THEOREM Let Ψ be a real statement, T a theory, and
F := Finitistic mathematics.
T proves Ψ
T`Ψ
=⇒
=⇒
F plus ConT proves Ψ
F + ConT ` Ψ.
Hilbert’s Consistency Proofs
• Grundlagen der Geometrie (1899). Shows the consistency of theories of geometries (euclidian and non-euclidian) by reduction to the theory of arithmetic.
• Über die Grundlagen der Logik und Arithmetik (1904) contains a consistency
proof of a weak theory of arithmetic (an almost equational theory).
• He shows that in this theory one can only deduce homogeneous equations,
hence no contradiction.
• Hilbert in lectures 1920,1921. New techniques for consistency proofs. The
ε-substitution method. Eliminates quantifiers.
Clear distinction between finitistic metatheory and object-theory.
Hilbert School I
• Wilhelm Ackermann (1896–1962): Begründung des tertium non datur mittels der Hilbertschen Theorie der Widerspuchsfreiheit (1925).
• Consistency proof for a theory of arithmetic with second order variables (ranging over functions). Function space closed under primitive recursion.
• The proof uses Hilbert’s ε-substitution. Very difficult to follow.
ω
• Proof seems to require a transfinite induction up to ω ω .
• John von Neumann (1903–1957) Zur Hilbertschen Beweistheorie (1927)
10
Hilbert School II
• Gerhard Gentzen (1909–1945)
• Untersuchungen über das logische Schliessen (1934) Dissertation:
• Introduces the natural deduction system and the sequent calculus. Proves cut
elimination.
• Die Widerspruchsfreiheit der reinen Zahlentheorie (1936)
• Proves the consistency of Peano arithmetic.
Herbrand
• Jacques Herbrand (1908–1931)
• Sur la non-contradiction de l’Arithmetique (1931)
The most important structure
• The set of natural numbers N = {0, 1, 2, 3, 4, . . .}
with operations of Addition (+) and Multiplication (×) and the less-than
relation (<):
N = (N; 0, 1, +, ×, <)
• Richard Dedekind (1831-1916), Giuseppe Peano (1858-1932)
Axiomatization of N: called Peano Arithmetic ( PA)
Usual laws for +, × and <.
• Axiom scheme of mathematical induction.
• Many of the famous theorems and problems of mathematics (including the
above examples) can be formalized as a sentence ϕ of the language of N and
thus are equivalent to the question whether N |= ϕ.
Is Ψ true in N?
Axiomatizing the Structure N Peano Arithmetic, PA.

 Predicate symbols : =, <
Function symbols : +, ·, S (Successor)
Language of PA :=

Constant symbols : 0
(N1) ∀x(Sx 6= 0)
(N2) ∀xy[Sx = Sy → x = y]
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(N3) ∀x[x + 0 = x]
(N4) ∀xy[x + Sy = S(x + y)]
(N5) ∀x[x · 0 = 0]
(N6) ∀xy[x · Sy = (x · y) + x]
(N7) ∀x¬(x < 0)
(N8) ∀xy[x < Sy ↔ x < y ∨ x = y]
(N9) ∀xy[x < y ∨ x = y ∨ y < x]
(IND) ϕ(0) ∧ ∀x[ϕ(x) → ϕ(Sx)] → ∀xϕ(x)
12
2
The sequent calculus
Remark: 2.1 The most common logical calculi are Hilbert-style systems. They
are specified by delineating a collection of schematic logical axioms and some inference rules. The choice of axioms and rules is more or less arbitrary, only subject to
the desire to obtain a complete system. In model theory it is usually enough to
know that there is a complete calculus for first order logic as this already entails the
compactness theorem.
There are, however, proof calculi without this arbitrariness of axioms and rules.
The natural deduction calculus and the sequent calculus were both invented
by Gentzen in 1934. Both calculi are pretty illustrations of the symmetries of logic.
In this course I shall focus on the sequent calculus since it is a central tool in ordinal
analysis and allows for generalizations to infinitary logics.
Gentzen’s main theorem about the sequent calculus is the Hauptsatz, i.e. cut
elimination.
2.1
Languages
As we will also consider intuitionistic theories and the intuitionistic version of the
sequent calculus it is in order to spell out what we consider to be the ingredients of
a first order theory.
Definition: 2.2 All first order languages will share the same logical symbols:
∧, ∨, →, ¬, ∀, ∃,
bound variables
x0 , x1 , x2 , x3 , . . .
and free variables
a0 , a1 , a2 , . . . .
A first order language L is specified by its non-logical symbols. These symbols are
separated into three groups: LC , LF , and LR . LC is the set of constant symbols,
LF is the set of function symbols, and LR is the set of relation symbols. Each
function symbol f ∈ LF also comes equipped with an arity #f which is a number
> 0. Likewise each relation symbol R ∈ LF comes equipped with an arity #R > 0.
The distinction between free and bound variables is not essential but it is extremely useful and simplifies arguments a great deal. Terms can be freely substituted for variables since variables occurring in them are always free and thus
cannot be captured by quantifiers. Also the cut elimination theorem to be proved
below would have to be reformulated in a slightly awkward way. For example,
P (x, y) → ∃y ∃x P (y, x) would not have a cut free proof.
Convention: 2.3 We will use metavariables x, y, z, u, v, . . . , y1 , y2 , . . . to range over
bound variables and a, b, c, d, b1 , b2 , b3 , . . . to range over free variables. We shall use
c, d, e, . . . , c0 , c1 , c2 , . . . to range over constants. Variables P, Q, R, S, R0 , R1 , R2 , . . . ,
will range over relation symbols while f, g, h, f0 , f1 , f2 , f3 , . . . , g0 , g1 , g2 , . . . range over
function symbols.
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Definition: 2.4 The terms of L are inductively defined as follows:
1. Every free variable is a term.
2. Every constant symbol (of L) is a term.
3. If f is an n-ary function symbol and s1 , . . . , sn are terms then f (s1 , . . . , sn ) is
a term.
Terms are often denoted by t, s, t1 , t2 , . . ..
The formulas of L are inductively defined as follows:
1. If R is an n-ary relation symbol of L and t1 , . . . , tn are terms the R(t1 , . . . , tn )
is a formula. R(t1 , . . . , tn ) is called an atomic formula.
2. If A and B are formulas, then so are (¬A), (A ∧ B), A ∨ B) and (A → B).
3. If A is a formula, a is a free variable and x is a bound variable not occurring
in A, then ∀x A0 and ∃x A0 are formulas, where A0 is the expression obtained
from A by replacing a everywhere in A by x.
Henceforth A, B, C, . . . , F, G, H, . . . will be metavariables ranging over formulas.
Definition: 2.5 A formula without free variables will be called a closed formula
or sentence.
In order to emphasize that they belong to a specific language L, a term or formula
of L will sometimes be called an L-term or L-formula.
To increase readability we shall omit parentheses whenever possible. Outer
parentheses will always be omitted. We shall observe the following priority rules: ¬
takes precedence over each of ∧ and ∨, and each of the latter two takes precedence
over →. For example, ¬A ∧ B is short for (¬A) ∨ B, and A ∧ B → A ∨ B is short for
(A ∧ B) → (A ∨ B). Parentheses will also be omitted in case of double negations:
e.g. ¬¬A stands for ¬(¬A). A ↔ B is short for (A → B) ∧ (B → A).
Convention: 2.6 If t is a term, we define the substitution of t for a free variable
a by A(t/a). To simplify notation, we adopt the convention that if A is a formula
and s is a term we often write A(s) to refer to the formula A with some (or even
no) occurrences of s in A indicated. If we then write A(t) afterwards in the same
context we refer to the result of replacing these indicated occurrences of s in A by
t.
We say that the variable a is fully indicated in A(a) if all occurrences of a in
A are indicated.
2.2
The rules
Definition: 2.7 A sequent (of L) is an expression Γ ⇒ ∆ where Γ and ∆ are
finite sequences of L-formulas A1 , . . . , An and B1 , . . . , Bm , respectively.
Γ ⇒ ∆ is read, informally, as Γ yields ∆ or, rather, the conjunction of the Ai
yields the disjunction of the Bj .
In particular,
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• If Γ is empty, the sequent asserts the disjunction of the Bj .
• If ∆ is empty, it asserts the negation of the conjunction of the Ai .
• if Γ and ∆ are both empty, it asserts the impossible, i.e. a contradiction.
We use upper case Greek letters Γ, ∆, Λ, Θ, Ξ . . . to range over finite sequences
of formulae.
Definition: 2.8 We spell out the axioms and the inference rules of the sequent
calculus.
Identity Axiom
A ⇒ A
where A is any formula. In point of fact, we shall limit this axiom to the case of
atomic formulae A.
CUT
Γ ⇒ ∆, A
A, Λ ⇒ Θ
Cut
Γ, Λ ⇒ ∆, Θ
A is called the cut formula of the inference.
Structural Rules
Exchange, Weakening, Contraction
Γ, A, B, Λ ⇒ ∆
Xl
Γ, B, A, Λ ⇒ ∆
Γ ⇒ ∆, A, B, Λ
Xr
Γ ⇒ ∆, B, A, Λ
Γ ⇒ ∆ W
l
Γ, A ⇒ ∆
Γ ⇒ ∆ W
r
Γ ⇒ ∆, A
Γ, A, A ⇒ ∆
Cl
Γ, A ⇒ ∆
Γ ⇒ ∆, A, A
Cr
Γ ⇒ ∆, A
LOGICAL INFERENCES
Negation
Γ ⇒ ∆, A
¬L
¬A, Γ ⇒ ∆
B, Γ ⇒ ∆
¬R
Γ ⇒ ∆, ¬B
Implication
Γ ⇒ ∆, A
B, Γ ⇒ Θ
→ L
A → B, Γ ⇒ ∆, Θ
A, Γ ⇒ ∆, B
→ R
Γ ⇒ ∆, A → B
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Conjunction
A, Γ ⇒ ∆
∧ L1
A ∧ B, Γ ⇒ ∆
B, Γ ⇒ ∆
∧ L2
A ∧ B, Γ ⇒ ∆
Γ ⇒ ∆, A
Γ ⇒ ∆, B
Γ ⇒ ∆, A ∧ B
∧R
Disjunction
A, Γ ⇒ ∆
B, Γ ⇒ ∆
∨L
A ∨ B, Γ ⇒ ∆
Γ ⇒ ∆, A
∨ R1
Γ ⇒ ∆, A ∨ B
Γ ⇒ ∆, B
∨ R2
Γ ⇒ ∆, A ∨ B
Quantifiers
F (t), Γ ⇒ ∆
∀L
∀x F (x), Γ ⇒ ∆
Γ ⇒ ∆, F (a)
∀R
Γ ⇒ ∆, ∀x F (x)
F (a), Γ ⇒ ∆
∃L
∃x F (x), Γ ⇒ ∆
Γ ⇒ ∆, F (t)
∃R
Γ ⇒ ∆, ∃x F (x)
In ∀L and ∃R, t is an arbitrary term. The variable a in ∀R and ∃L is an eigenvariable
of the respective inference, i.e. a is not to occur in the lower sequent.
Definition: 2.9 The formulae in a logical inference marked blue are called the
minor formulae of that inference, while the red formula is the principal formula of
that inference. The other formulae of an inference are called side formulae.
A proof (aka deduction or derivation) D is a tree of sequents satisfying the following
conditions:
• The topmost sequents of D are identity axioms.
• Every sequent in D except the lowest one is an upper sequent of an inference
whose lower sequent is also in D.
Definition: 2.10 (The INTUITIONISTIC case.) The intuitionistic sequent
calculus is obtained by requiring that all sequents be intuitionistic. A sequent
Γ ⇒ ∆ is said to be intuitionistic if ∆ consists of at most one formula.
Specifically, in the intuitionistic sequent calculus there are no inferences corresponding to contraction right or exchange right.
16
Our first example is a deduction of the law of excluded middle.
A ⇒ A
¬R
⇒ A, ¬A
∨R
⇒ A, A ∨ ¬A
Xr
⇒ A ∨ ¬A, A
∨R
⇒ A ∨ ¬A, A ∨ ¬A
Cr
⇒ A ∨ ¬A
Notice that the above proof is not intuitionistic since it involves sequents that are
not intuitionistic.
The second example is an intuitionistic deduction.
F (a) ⇒ F (a)
∃R
F (a) ⇒ ∃x F (x)
¬L
¬∃x F (x), F (a) ⇒
Xl
F (a), ¬∃x F (x) ⇒
¬L
¬∃xF (x) ⇒ ¬F (a)
∀R
¬∃x F (x) ⇒ ∀x ¬F (x)
→R
⇒ ¬∃x F (x) → ∀x ¬F (x)
Convention: 2.11 Logics without (some of the) structural rules became important
in the 1980s. In particular Linear Logic attracted a great deal of attention back
then. For our purposes the structural rules just add an additional layer of bureaucracy. We would really like to sweep them under the carpet. We will achieve this by
identifying a sequence of formulas A1 , . . . , An with the set of formulas {A1 , . . . , An }.
Henceforth variables ∆, Γ, Λ, . . . will range over finite sets of formulas. We will interpret a comma between these sets as set-theoretic union. Thus Γ, ∆ stands for Γ ∪ ∆.
We also adopt the convention that Γ, A stands for Γ ∪ {A}. Likewise A1 , . . . , An
stands for {A1 , . . . , An } and Γ, ∆, A stands for Γ ∪ ∆ ∪ {A} etc.
Since in the curly bracket notation {A1 , . . . , An } the ordering of the formulas
does not matter and repeating a formula doesn’t make a difference, this will take
care of the exchange and the contraction rules automatically.
This still leaves the weakening rules. However, we are going to ditch them
completely in the classical case since it is always possible to add more side formulas
already at the leaves of a proof tree. Thus we adopt as Axioms all sequents of the
form
Γ, A ⇒ ∆, A
where A is an atomic formula. Thus, henceforth we no longer consider explicit
structural rules in the classical case.
The left rule for → can be simplified a bit in the classical case. Henceforth we
adopt this rule:
Γ ⇒ ∆, A
B, Γ ⇒ ∆
→ L
A → B, Γ ⇒ ∆
17
while the intuitionistic rule takes the form
Γ ⇒ A
B, Γ ⇒ ∆
→ L
A → B, Γ ⇒ ∆
with ∆ containing at most one formula.
In the intuitionistic case, we shall also ditch the structural rules with one exception. Here the Axioms will be all the sequents of the form
∆, A ⇒ A
with A atomic. As a result we no longer need the left weakening rule. However we
still need the right weakening rule that is from
Γ ⇒
we may infer
Γ ⇒ B
for any formula B. This rule could also be called ex falso quodlibet.
Definition: 2.12 A sequent deduction D is a proof tree and we can measure a tree
by its height, i.e. its longest branch. We use |D| to denote the height of D.
We shall use the notation
Γ ⇒ ∆ to express that there is a deduction of
Γ ⇒ ∆ while
n
Γ ⇒ ∆
is used to convey that there is a deduction of Γ ⇒ ∆ with height ≤ n.
We use
n
I Γ ⇒ ∆
to convey that that there is a deduction of Γ ⇒ ∆ with height ≤ n in the intuitionistic sequent calculus, and I Γ ⇒ ∆ to say that there is an intutitionistic
deduction.
The length |A| of a formula A is defined as follows: |A| = 0 if A is atomic.
|¬A| = |A| + 1, |A♦B| = max(|A|, |B|) + 1 if ♦ is one of the connectives ∨, ∧, →,
|∃x A| = |A| + 1, |∀x A| = |A| + 1.
We write
n
Γ ⇒ ∆
k
if there is a deduction of Γ ⇒ ∆ of height ≤ n such that all cuts in this deduction
have cut formulas with length < k.
n
I k Γ ⇒ ∆ is defined similarly.
Lemma: 2.13 For every formula A there is an intuitionistic deduction of A ⇒ A.
t
u
Proof: Exercise.
We list some technical lemmata that will be useful for proving cut elimination.
Lemma: 2.14 (Substitution) Let Γ(a) and ∆(a) be sets of formulas with all occurrences of a indicated. Let s be an arbitrary term.
18
(i) If
(ii) If I
n
k
n
k
n
Γ(a) ⇒ ∆(a) , then
k
Γ(a) ⇒ ∆(a) , then I
Lemma: 2.15 (Weakening)
(ii) If I
n
k
Γ ⇒ ∆ , then I
n
k
Γ(s) ⇒ ∆(s) .
n
k
Γ(s) ⇒ ∆(s) .
n
(i) If
k
Γ ⇒ ∆ , then
n
k
Γ, Γ0 ⇒ ∆, ∆0 .
Γ, Γ0 ⇒ ∆ .
Proof: Just add Γ0 and ∆0 to all sequents in the deduction. Formally one proves
this by induction on n. In the cases of quantifier rules with eigenvariable conditions
one might have to replace these variables by ‘fresh’ ones, using Lemma 2.14.
t
u
Lemma: 2.16 (Inversion)
(ii) If
(iii) If
(iv) If
(v) If
(vi) If
(vii) If
(viii) If
(ix) If
(x) If
n
k
n
k
n
k
n
k
n
k
n
k
n
k
n
k
n
k
(i) If
n
Γ ⇒ ∆, A ∧ B then
n
n
Γ, A → B ⇒ ∆ then
Γ ⇒ ¬A, ∆ then
Γ, ¬A ⇒ ∆ then
n
k
n
k
n
k
n
k
n
k
Γ, A, B ⇒ ∆ .
Γ ⇒ ∆, B .
Γ, B ⇒ ∆ .
Γ ⇒ ∆, A, B .
k
Γ ⇒ A → B, ∆ then
Γ, A ∧ B ⇒ ∆ then
Γ, A ⇒ ∆ and
k
Γ ⇒ ∆, A ∨ B then
k
Γ ⇒ ∆, A and
k
Γ, A ∨ B ⇒ ∆ then
n
n
k
n
k
A, Γ ⇒ ∆, B .
Γ ⇒ ∆, A and
n
k
Γ, B ⇒ ∆ .
Γ, A ⇒ ∆ .
Γ ⇒ ∆, A .
Γ ⇒ ∆, ∀x B(x) then
Γ, ∃x B(x) ⇒ ∆ then
n
k
n
k
Γ ⇒ ∆, B(s) for any term s.
Γ, B(s) ⇒ ∆ for any term s.
(xi) With the exception of (iv), (vi) and (viii) the above inversion properties remain
valid for the intuitionistic sequent calculus. One half of (vi) also remains valid
intutionistically:
If I
n
k
Γ, A → B ⇒ ∆ then I
n
k
Γ, B ⇒ ∆ .
Proof: All are provable by easy inductions on n.
We have laid the groundwork for cut elimination.
Here is an example of how to eliminate cuts of a special form:
A, Γ ⇒ ∆, B
Λ ⇒ Θ, A
B, Ξ ⇒ Φ
→R
→L
Γ ⇒ ∆, A → B
A → B, Λ, Ξ ⇒ Θ, Φ
Cut
Γ, Λ, Ξ ⇒ ∆, Θ, Φ
is replaced by
Λ ⇒ Θ, A
A, Γ ⇒ ∆, B
Cut
Λ, Γ ⇒ Θ, ∆, B
B, Ξ ⇒ Φ
Cut
Γ, Λ, Ξ ⇒ ∆, Θ, Φ
19
t
u
So we have replaced a cut with cut formula A → B by cuts with formulas of smaller
length. By doing this systematically we arrive at the Reduction Lemma. Well,
actually it is not that easy when contractions are involved, i.e. when the principal
formula of an inference is also a side formula:
A, Γ ⇒ ∆, B, A → B
Λ, A → B ⇒ Θ, A
B, Ξ, A → B ⇒ Φ
→R
→L
Γ ⇒ ∆, A → B
A → B, Λ, Ξ ⇒ Θ, Φ
Cut
Γ, Λ, Ξ ⇒ ∆, Θ, Φ
n
Lemma: 2.17 (Reduction) Suppose k ≤ |C|. If k Γ, C ⇒ ∆ and
then
2(n+m)
Γ, Ξ ⇒ ∆, Θ .
|C|
m
k
Ξ ⇒ Θ, C ,
Proof: Of course we could derive Γ, Ξ ⇒ ∆, Θ by an application of the cut rule,
but the resulting derivation would have cut rank |C| + 1.
The proof is by induction on n + m. Let D1 be a derivation of Γ, C ⇒ ∆ with
cut rank ≤ k and length ≤ n. Likewise let D2 be a derivation of Ξ ⇒ C, Θ with
cut rank ≤ k and length ≤ m.
Case 1: Γ, C ⇒ ∆ is an axiom whose principal formula is not C, i.e., Γ = Γ0 , A
and ∆ = ∆0 , A for some atom A. Then Γ, Ξ ⇒ ∆, Θ is an axiom too and the desired
assertion follows.
Similarly, if Ξ ⇒ Θ, C is an axiom whose principal formula is different from C
then Ξ ⇒ Θ is an axiom and so is Γ, Ξ ⇒ ∆, Θ.
Case 2: Both Γ, C ⇒ ∆ and Ξ ⇒ Θ, C are axioms with principal formula C.
Then ∆ = ∆0 , C and Ξ = Ξ0 , C for some ∆0 and Ξ0 . Hence Γ, Ξ ⇒ ∆, Θ is an axiom
as well.
Henceforth we may assume that Γ, C ⇒ ∆ or Ξ ⇒ Θ, C is not an axiom. Hence
at least one of the derivations ends with an inference which will be called its last
inference.
Case 3: D1 ends with an inference whose principal formula is different from C.
Then the premisses of the last inference are of the form
Γi , C ⇒ ∆i
ni
and we have k Γi , C ⇒ ∆i where ni < n. Since ni + m < n + m we can apply the
induction hypothesis to the premisses and obtain
2(ni +m)
|C|
Γi , Ξ ⇒ ∆i , Θ .
2(n+m)
Γ, Ξ ⇒ ∆, Θ . If the last inference
By applying the same inference we get
|C|
comes with an eigenvariable condition it might be necessary to substitute a new
variable. But by Lemma 2.14 this can be done without increasing length and cut
rank of derivations.
Case 4: D2 ends with an inference whose principal formula is different from C.
This is analogous to the previous case.
We may from now on assume that C is the principal formula of the last inference of
20
both D1 and D2 . In particular C is not an atom.
Case 5: C is of the form A ∧ B. Then we have
n1
k
Γ, C, A ⇒ ∆
(1)
Γ, C, B ⇒ ∆
(2)
Ξ ⇒ Θ, C, A
(3)
Ξ ⇒ Θ, C, B
(4)
or
n1
k
as well as
m1
k
and
m2
k
for some n1 < n and m1 , m2 < m. Note that C could have been a side formula of
any of the last inferences of D1 and D2 , and, moreover, that by weakening (Lemma
2.15) we can always add C as a side formula without increasing the length or the
cut rank of the derivation.
m
If (1) obtains we apply the induction hypothesis with (1) and k Ξ ⇒ Θ, C to
arrive at
2(n1 +m)
Γ, Ξ, A ⇒ ∆, Θ .
|C|
(5)
Applying the Inversion Lemma 2.16 (ii) to (3) we have
m1
k
Ξ ⇒ Θ, A .
(6)
Cutting A out of (5) and (6) gives the desired
2(n+m)
|C|
Γ, Ξ ⇒ ∆, Θ
since |A| < |C|.
If (2) obtains we apply the induction hypothesis with (2) and
arrive at
2(n1 +m)
|C|
m
k
Γ, Ξ, B ⇒ ∆, Θ .
Ξ ⇒ Θ, C to
(7)
Applying the Inversion Lemma 2.16 (ii) to (4) we have
m1
k
Ξ ⇒ Θ, B .
Cutting B out of (7) and (8) gives the desired
2(n+m)
|C|
(8)
Γ, Ξ ⇒ ∆, Θ .
Case 6: C is of the form ∀x A(x). Then we have
n1
k
Γ, C, A(s) ⇒ ∆
21
(9)
and
m1
k
Ξ ⇒ Θ, C, A(a)
(10)
for some n1 < n and m1 < m with a being an eigenvariable. Applying the induction
m
hypothesis to (9) and k Ξ ⇒ Θ, C we get
2(n1 +m)
|C|
Γ, Ξ, A(s) ⇒ ∆, Θ .
(11)
By applying first inversion (Lemma 2.16) to (10) and subsequently substitution
(Lemma 2.14) (or the other way round) we get
m1
Ξ ⇒ Θ, A(s) .
k
A cut performed on (11) and (12) yields
2(n+m)
|C|
(12)
Γ, Ξ ⇒ ∆, Θ .
Case 7: C is of the form A → B. Then we have
n1
Γ, C ⇒ ∆, A
(13)
Γ, C, B ⇒ ∆
(14)
Ξ, A ⇒ Θ, C, B .
(15)
k
and
n2
k
as well as
m1
k
for some n1 , n2 < n and m1 < m.
m
(13) can be linked up with k Ξ ⇒ Θ, C to furnish a pair to which we can apply
the induction hypothesis. Whence we get
2(n1 +m)
|C|
Γ, Ξ ⇒ ∆, Θ, A .
(16)
Another pair to which we can apply the induction hypothesis is given by (15) and
m
Γ, C ⇒ ∆ . Thus
k
2(n+m1 )
|C|
Γ, Ξ, A ⇒ ∆, Θ, B .
(17)
Applying a cut to (17) and (16) yields
max(2(n+m1 ),2(n1 +m))+1
|C|
Γ, Ξ ⇒ ∆, Θ, B .
(18)
Applying the Inversion Lemma 2.16 (xi) to (14) yields
n1
k
Γ, B ⇒ ∆ .
(19)
Cutting out B from (18) and (19) we arrive at
max(2(n+m1 ),2(n1 +m))+2
|C|
22
Γ, Ξ ⇒ ∆, Θ .
(20)
As max(2(n + m1 ), 2(n1 + m)) + 2 ≤ 2(n + m) we get the desired result from (20).
Case 8: C is of the form A ∨ B. Then we have
n1
k
Γ, C, A ⇒ ∆
(21)
Γ, C, B ⇒ ∆
(22)
Ξ ⇒ Θ, C, A
(23)
Ξ ⇒ Θ, C, B
(24)
and
n2
k
and also
m1
k
or
m1
k
for some n1 , n2 < n and m1 < m. To (21) and
hypothesis to arrive at
2(n1 +m)
|C|
To (22) and
m
k
m
k
Ξ ⇒ Θ, C we apply the induction
Γ, Ξ, A ⇒ ∆, Θ .
(25)
Ξ ⇒ Θ, C we apply the induction hypothesis to arrive at
2(n2 +m)
|C|
Γ, Ξ, B ⇒ ∆, Θ .
(26)
From (23) as well as (24) we get
m1
k
Ξ ⇒ Θ, A, B
(27)
by the Inversion Lemma 2.16 (iv). Cutting A out of (25) and (27) yields
2(n1 +m)+1
|C|
Γ, Ξ ⇒ ∆, Θ, B .
(28)
Performing a cut on (26) and (28) gives
2(n+m)
|C|
Γ, Ξ ⇒ ∆, Θ .
Case 9: C is of the form ∃x A(x). Then we have
n1
k
Γ, C, A(a) ⇒ ∆
(29)
Ξ ⇒ Θ, C, A(s)
(30)
and
m1
k
23
for some n1 < n and m1 < m with a being an eigenvariable. Applying the induction
n
hypothesis with (30) and k Γ, C ⇒ ∆ we get
2(n+m1 )
|C|
Γ, Ξ ⇒ ∆, Θ, A(s) .
(31)
By applying first inversion (Lemma 2.16) to (29) and subsequently substitution
(Lemma 2.14) (or the the other way round) we get
n1
k
Γ, A(s) ⇒ Θ .
2(n+m)
A cut performed on (31) and (32) yields
|C|
(32)
Γ, Ξ ⇒ ∆, Θ .
Case 10: C is of the form ¬A. Then we have
n1
k
Γ, C ⇒ ∆, A
(33)
Ξ, A ⇒ Θ, C .
(34)
and
m1
k
for some n1 < n and m1 < m. The induction hypothesis applies to (33) and
m
Ξ ⇒ Θ, C , furnishing
k
2(n1 +m)
|C|
Γ, Ξ ⇒ ∆, Θ, A .
(35)
Now apply the Inversion Lemma 2.16 (vii) to (34) to get
m1
k
Ξ, A ⇒ Θ .
(36)
Cutting out A from (35) and (36) we arrive at
2(n+m)
|C|
Γ, Ξ ⇒ ∆, Θ .
t
u
Theorem: 2.18 (Cut Reduction) If
n
Γ ⇒ ∆ then
k+1
4n
k
Γ ⇒ ∆.
Proof: We use induction on n. Suppose D is a derivation of Γ ⇒ ∆ with length
≤ n and cut rank ≤ k + 1. If Γ ⇒ ∆ is an axiom then we clearly get the desired
result. So let’s assume that Γ ⇒ ∆ is not an axiom. Then D has a last inference (I)
with premisses Γi ⇒ ∆i . Suppose the inference was not a cut or a cut of a degree
ni
< k. We then have k Γi ⇒ ∆i for some ni < n. By the induction hypothesis
4ni
4n
we have k Γi ⇒ ∆i . Applying the same inference (I) yields k Γ ⇒ ∆ since
4ni < 4n .
Now suppose the last inference was a cut with a cut formula C satisfying |C| = k.
By the induction hypothesis we have
4n1
k
Γ, C ⇒ ∆
24
and
4n2
k
Γ ⇒ ∆, C
for some n1 , n2 < n. We can then apply the Reduction Lemma 2.17 to these deriva2(4n1 +4n2 )
tions and arrive at k
Γ ⇒ ∆ . Since 2(4n1 +4n2 ) ≤ 4n the desired conclusion
follows.
t
u
m
m
4r
Corollary: 2.19 (Gentzen’s Hauptsatz) Let 4m
0 = m and 4r+1 = 4 .
If
n
k
Γ ⇒ ∆ then
4n
k
0
Γ ⇒ ∆.
As a result, there is a cut free derivation of Γ ⇒ ∆.
Proof: Just apply the previous result k times. Formally that is an induction on
k.
t
u
Definition: 2.20 For a formula A we define its set of subformulae, Subf(A) as
follows: If A is an atom then Subf(A) = {A}. Subf(¬A) = Subf(A) ∪ {¬A}.
Subf(A♦B) = Subf(A) ∪ Subf(B) ∪ {A♦B} if ♦ is one of the connectives ∧, ∨, →.
[
Subf(Qx F (x)) = {Qx F (x)} ∪
Subf(F (s))
s∈T erm
where Q is ∀ or ∃ and T erm is the set of terms.
B is said to be a subformula of A if B ∈ Subf(A).
Corollary: 2.21 (The subformula property) The Hauptsatz 2.19 has an important corollary.
If a sequent Γ ⇒ ∆ is deducible, then it has a deduction such that every formula
occurring in it is a subformula of some formula in γ ∪ ∆.
Proof: Take a cut free proof of Γ ⇒ ∆. Then it’s clear the the entire deduction is
made of subformulas of formulas in Γ and ∆.
t
u
Corollary: 2.22 A contradiction, i.e. the empty sequent cannot be deduced.
Proof: The empty sequent cannot have a cut free deduction. What could have
been the last inference?
t
u
2.3
Cut elimination for the intuitionistic sequent calculus
Lemma: 2.23 (Reduction) Suppose k ≤ |C|. If I
then
2(n+m)
Γ, Ξ ⇒ ∆ .
I
n
k
Γ, C ⇒ ∆ and I
m
k
Ξ ⇒ C,
|C|
t
u
Proof: The proof is similar to the classical case (Lemma 2.17).
m
m
4r
Corollary: 2.24 (Gentzen’s Hauptsatz) Let 4m
0 = m and 4r+1 = 4 .
If I
n
k
Γ ⇒ ∆ then I
4n
k
0
Γ ⇒ ∆.
As a result, there is a cut free intuitionistic derivation of Γ ⇒ ∆.
25
3
Consequences of the Hauptsatz
Definition: 3.1 A formula is said to be existential if it is quantifier free or of the
form ∃x1 . . . , ∃xr B(x1 , . . . , xr ) with B(a1 , . . . , br ) quantifier free.
Note that a subformula of an existential formula is existential too.
Lemma: 3.2 Suppose that Γ consists of quantifier free formulae and ∆ consists
entirely of existential formulae. Let ∃x C(x) be an existential formula. If
Γ ⇒ ∆, ∃x C(x)
then there exist terms t1 , . . . , tk such that
Γ ⇒ ∆, C(t1 ), . . . , C(tk ) .
Proof: By the Hauptsatz we have a cut free deduction D of Γ ⇒ ∆, ∃x C(x). We
proceed by induction on n = |D|. If n = 0 then Γ ⇒ ∆ is already an axiom. Now
let n > 0. The D ended with an inference. First suppose the last inference of D does
not have ∃x C(x) as principal formula. Then its premisses are of the form Γi ⇒
∆i , ∃x C(x). Note that the formulae of Γi must also be quantifier free and those in
∆i must be existential too. Let’s assume we have two premisses. Inductively we
Γi ⇒ ∆i , C(ti1 ), . . . , C(tiri ) for some terms and by applying weakening
then have
and the same inference we get
Γ ⇒ ∆, C(t11 ), . . . , C(t1r1 ), C(t21 ), . . . , C(t2r2 ) .
If ∃x C(x) is the principal formula of the last inference of D then this must have
been ∃R and its premiss is of the form Γ ⇒ ∆, ∃x C(x), C(t) for some term t.
Inductively we have terms t01 , . . . , t0l such that
Γ ⇒ ∆, C(t01 ), . . . , C(t0l ), C(t) and
we are done.
t
u
We shall sometimes write ` ∆ and I ` ∆ for
tively.
⇒ ∆ and I
⇒ ∆ , respec-
Theorem: 3.3 (Herbrand’s Theorem) If A(~a, ~b ) is quantifier free and
` ∀~x ∃~y A(~x, ~y )
then there are finitely many term tuples t1 , . . . , tn each of the same length as ~b whose
free variables are among ~a such that
` A(~a, t1 ) ∨ . . . ∨ A(~a, tn ).
Proof: Using inversion 2.16 (ix) several times we have ` ∃~y A(~a, ~y ). Now use
Lemma 3.2 several times followed by several ∨R inferences.
t
u
The intuitionistic case is much easier to prove.
Lemma: 3.4 If I
∃y F (y) then I
F (t) for some term t.
26
Proof: We have I
n
0
∃y F (y) for some n. The last inference of the pertaining deducn−1
tion must have been ∃R. Hence I 0 F (t) for some term t since in the intuitionistic
case we can not have side formulas in the antecedent.
t
u
Corollary: 3.5 If
I
∀~x ∃~y A(~x, ~y )
then there exists a term tuple t of the same length as ~b whose free variables are
among ~a such that
I A(~a, t) .
Proof: Use ∀-inversion and apply the previous Lemma several times.
t
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Examples: 3.6 In the classical case we cannot always find a single term as the
following example demonstrates. Let L be a language that has two constants 0, 1
and two unary predicate symbols P and R. Then in classical logic we have
` ∃y [(P (0) → R(0)) ∧ (¬P (0) → R(1)) → R(y)]
but we can not prove
(P (0) → R(0)) ∧ (¬P (0) → R(1)) → R(t)
for any term t. (Exercise)
Definition: 3.7 A theory T is a set of sentences, called its axioms. T is said to be
universal (or open) if all of its axioms are of the form ∀~x A(~x ) with A(~a ) quantifier
free.
If ~s is a tuple of terms (of the same length as ~x) then A(~s ) will be called a
substitution instance of ∀~x A(~x ).
Theorem: 3.8 (Hilbert-Ackermann Consistency Theorem) A universal theory T is inconsistent iff there is a tautology which is a disjunction of negations of
substitution instances of the axioms of T . In other words T is inconsistent iff there
are substitution instances B1 , . . . , Bn of axioms of T such that
` ¬B1 ∨ . . . ∨ ¬Bn .
Proof: Clearly if ` ¬B1 ∨ . . . ∨ ¬Bn holds then T must be inconsistent since T
proves each Bi . Conversely, if T is inconsistent then there are finitely many axioms
A1 , . . . , An of T such that
A1 , . . . , An ⇒ .
(37)
Each Ai is of the form ∀~x Ci (~x ) with Ci (~a ) quantifier free. By applying ¬R to (37)
n times we obtain
⇒ ¬A1 , . . . , ¬An .
(38)
Since ` ¬Ai ⇒ ∃~x ¬Ci (~x ) holds for all i we can employ n cuts to (38) to arrive at
⇒ ∃~x ¬C1 (~x ), . . . , ∃~x ¬Cn (~x ).
(39)
Now apply Lemma 3.2 to (39) several times to get rid of the existential quantifiers
and subsequently apply ∨R several times to get the desired result.
t
u
27
Remark: 3.9 There are many examples of universal theories: the theory of equality,
the theory of groups with a constant symbol for the neutral element and a function
symbol for the inverse operation, the theory of linear orderings and many equational
theories.
Next we will turn to a richer class of theories, the so-called geometric theories.
Definition: 3.10 The geometric formulae are inductively defined as follows: Every atom is a geometric formula. If A and B are geometric formulae then so are
A ∨ B, A ∧ B and ∃x A.
Another way of saying this is that a formula is geometric iff it does not contain
any of the particles →, ¬, ∀.
A formula is called a geometric implication if it is of either form ∀~x A or ∀~x ¬A
or ∀~x (A → B) with A and B being geometric formulae. Here ∀~x may be empty. In
particular geometric formulae and their negations are geometric implications.
A theory is geometric if all its axioms are geometric implications.
Examples: 3.11 (i) 1. Robinson arithmetic. The language has a constant 0, a
unary successor function suc and binary functions + and ·. Axioms are the
equality axioms and the universal closures of the following.
1. ¬suc(a) = 0.
2. suc(a) = suc(b) → a = b.
3. a = 0 ∨ ∃y a = suc(y).
4. a + 0 = a.
5. a + suc(b) = suc(a + b).
6. a · 0 = 0.
7. a · suc(b) = a · b + a
A classically equivalent axiomatization is obtained if (3) is replaced by
¬a = 0 → ∃y a = suc(y)
but this is not a geometric implication.
(ii) The theories of groups, rings, and local rings have geometric axiomatizations.
(iii) The theories of fields, ordered fields, algebraically closed fields and real closed
fields have geometric axiomatizations.
To express algebraic closure replace axioms
s 6= 0 → ∃x sxn + t1 xn−1 + . . . + tn−1 x + tn = 0
by
s = 0 ∨ ∃x sxn + t1 xn−1 + . . . + tn−1 x + tn = 0
where sxk is short for s · x · . . . · x with k many x.
(iv) The theory of projective geometry has a geometric axiomatization.
28
We want to show that a geometric implication which is classically deducible in a
geometric theory T is also intuitionistically deducible in T . We need some simple
observations.
Lemma: 3.12 Let π {1, . . . , n} → {1, . . . , n} be a bijection.
1. If I
Γ ⇒ A1 ∨ . . . ∨ An then I
Γ ⇒ Aπ(1) ∨ . . . ∨ Aπ(n) .
2. If I
Γ ⇒ D ∨ F (s) then I
3. If I
Γ ⇒ D ∨ B and I
Γ ⇒ D ∨ C then I
4. If I
Γ ⇒ A ∨ B then I
Γ, ¬A ⇒ B .
5. If I
Γ, B ⇒ C and I
Γ ⇒ D ∨ ∃x F (x) .
Γ ⇒ C ∨ A then I
Γ ⇒ D ∨ (B ∧ C) .
Γ, A → B ⇒ C .
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Proof: Exercise.
W
Lemma: 3.13 For a finite set of formulas
∆
=
{A
,
.
.
.
,
A
}
let
∆ be the formula
1
n
W
A1 ∨ . . . ∨ An . If ∆ is empty then ∆ is the empty set.
Let Γ be a finite set of geometric implications and ∆ be a finite set of geometric
formulas.
W
Γ ⇒ ∆ then I Γ ⇒ ∆ .
If
Proof: Let D be a cut free deduction of Γ ⇒ ∆. The proof proceeds by induction
on n = |D|. If Γ ⇒ ∆ is an axiom then there exists an atom A such that A ∈ Γ ∩ ∆.
If ∆ has no other formulae we are done. If there are other
formulae in ∆, say
W
D1 , . . . , Dk , then apply ∨R k times to arrive at I Γ ⇒ ∆ .
Let n > 0. We inspect the last inference of D. Note that ∀R, ¬R and → R are
ruled out since they have non-geometric principal formulas. If the last inference was
∀L, ∃L, ∧L, or ∨L we can simply apply the induction hypothesis to the premisses
and re-apply the same inference.
If the last inference was ∃R apply the induction hypothesis to its premiss and
subsequently use Lemma 3.12 (2) to get the desired result.
If the last inference was ∧R apply the induction hypothesis to its premisses and
subsequently use Lemma 3.12 (3).
If the last inference was ¬L then its minor formula must be geometric. Then
apply the induction hypothesis to its premiss and subsequently use Lemma 3.12 (4).
If the last inference was → L then apply the induction hypothesis to its premisses
and subsequently use Lemma 3.12 (5).
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Theorem: 3.14 Let T be a geometric theory and suppose that there is a classical
proof of a geometric implication G in T . Then there is an intuitionistic proof of G
from the axioms of T .
Proof: G is of the form ∀~x F (~x ) where F (~a ) is a geometric formula or the negation
of a geometric formula or an implication of two geometric formulae.
We have
A1 , . . . , Ak ⇒ G
29
for some axioms A1 , . . . , Ak of T . Using the Inversion Lemma 2.16 (ix) we get
A1 , . . . , Ak ⇒ F (~a ) .
If F (~a ) is geometric we obtain I A1 , . . . , Ak ⇒ F (~a ) by Lemma 3.13 so that via
(several) ∀R inferences we arrive at the desired result.
If F (~a ) is of the form ¬F0 (~a ) with F0 (~a ) geometric we apply the Inversion
Lemma 2.16 (vii) to get
A1 , . . . , Ak , F0 (~a ) ⇒ .
By Lemma 3.13 we infer that I
I
A1 , . . . , Ak , F0 (~a ) ⇒ and thus, by ¬R, we have
A1 , . . . , Ak ⇒ ¬F0 (~a )
so that via ∀R we arrive at I A1 , . . . , Ak ⇒ ∀~x ¬F0 (~x ) .
If F (~a ) is of the form F0 (~a ) → F1 (~a ) with Fi (~a ) geometric we apply the Inversion
Lemma 2.16 (v) to get
A1 , . . . , Ak , F0 (~a ) ⇒ F1 (~a ) .
By Lemma 3.13 we infer that I A1 , . . . , Ak , F0 (~a ) ⇒ F1 (~a ) . Via → R we get
I A1 , . . . , Ak ⇒ F0 (~a ) → F1 (~a ) and via ∀R we arrive at
I
A1 , . . . , Ak ⇒ ∀~x (F0 (~x ) → F1 (~x )) .
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The previous result
W can be extended toVinfinitary languages which accommodate
infinite disjunctions Φ and conjunctions Φ, where Φ is set of (infinitary) formulas
such that the total number of variables (free and bounded) occurring in the formulas
of Φ is finite. In this richer language a formula
W is said to be coherent if in addition to
∨, ∧, ∃ one also allows infinite disjunctions Φ, where Φ is already a set of coherent
formulas satisfying the above proviso on the number of variables. Then a theorem
similar to 3.14 can be shown for coherent theories, that is theories axiomatized by
coherent implications.
An example of an axiom expressible in this richer language via a coherent implication is the Archimedian axiom:
∀x (x < 1 ∨ x < 1 + 1 ∨ . . . ∨ x < 1 + . . . + 1 ∨ . . .)
or in more compact way:
∀x
_
x < n.
n∈N
Geometric theories are quite ubiquitous. There exists a simple method which
is sometimes called Morleyisation (in honour of the logician Michael Morley) by
which every theory can be given a geometric axiomatization in a richer language.
The technique actually goes back to Skolem. Albeit Skolemization would be more
appropriate that name is already used for something else. Wilfrid Hodges called the
procedure to find a ∀∃ axiomatization in a richer language atomization.
30
Definition: 3.15 Below ∀~x (A1 (~x ) A2 (~x )) will stand for two formulas namely
∀~x (A1 (~x ) → A2 (~x )) and ∀~x (A2 (~x ) → A1 (~x )).
Let T be a theory in a first order language L. For each formula A(a1 , . . . , an ) of
L with all free variables indicated we add two new n-ary relation symbols PA(~a ) and
NA(~a ) to the language, where ~a = a1 , . . . , an . Call the new language La . The theory
T a in the language La has the following axioms:
1. ∀~x ¬(PA(~a ) (~x) ∧ NA(~a ) (~x)).
2. ∀~x (PA(~a ) (~x) ∨ NA(~a ) (~x)).
3. If A(~a ) is atomic add the axioms ∀~x (PA(~a ) (~x) A(~x )).
4. If A(~a ) is B(~a ) ∧ C(~a ) add ∀~x (PA(~a ) (~x ) PB(~a ) (~x ) ∧ PC(~a ) (~x )).
5. If A(~a ) is B(~a ) ∨ C(~a ) add ∀~x (PA(~a ) (~x ) PB(~a ) (~x ) ∨ PC(~a ) (~x )).
6. If A(~a ) is ¬B(~a ) add ∀~x (PA(~a ) (~x ) NB(~a) (~x )).
7. If A(~a ) is B(~a ) → C(~a ) add ∀~x (PA(~a ) (~x ) NB(~a ) (~x ) ∨ PC(~a ) (~x )).
8. If A(~a ) is ∃yB(~a, y) add ∀~x (PA(~a ) (~x ) ∃y PB(~a,b) (~x, y)).
9. If A(~a ) is ∀yB(~a, y) add ∀~x (NA(~a ) (~x ) ∃y NB(~a,b) (~x, y)).
10. Finally, for each axiom ∀~x A(~x ) of T add ∀~x PA(~a) (~x ) as an axiom to T a .
Clearly T a is a geometric theory.
Theorem: 3.16 Let T and T a as above.
(i) For every formula A(~a ) of L with all free variables indicated,
T a ` ∀~x [A(~x ) ↔ PA(~a) (~x )].
(ii) Every model A of T can be expanded in just one way to an La -structure Aa
which is a model of T a .
(iii) T a is conservative over T , that is, for every L-sentence B,
T ` B iff T a ` B.
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Proof: Exercise.
31
4
Ordinal functions and representation systems
The strength of appropriate theories can be aptly measured via transfinite ordinals.
To be able to denote these ordinals and have a sufficient supply of them we shall
go beyond the operations of addition, multiplication and exponentiation on ordinals
and study a hierarchy of functions introduced by O. Veblen in 1908. In what follows
we will work informally in a sufficiently strong classical set theory, e.g. ZF. Lower
case Greek letters α, β, γ, δ, . . . will be assumed to range over the class of ordinals
ON. 0 is the smallest ordinal. Every ordinal α has a successor which we denote by
α + 1, i.e., α + 1 is the smallest ordinal that is bigger than α. An ordinal of the form
α + 1 is a successor ordinal or just a successor. A limit ordinal or just a limit
is an ordinal which is not a successor and > 0. We denote the ordering of ordinals
by < and the less-than-or-equal relation by ≤.
As per usual we identify an ordinal α with the set {β | β < α}. In set theory an
ordinal is defined to be a transitive set whose elements are transitive too. Moreover,
< on ordinals coincides with ∈ and thus α = {β | β ∈ α}.
Some crucial properties about ordinals that we shall assume are the following.
Postulates: 4.1 (Ordinals)
(O1) < is a total linear ordering on ON, i.e. α 6< α and α < β ∨ β < α ∨ α = β
hold for all α and β.
(O2) Every non-empty class X of ordinals contains a least element (necessarily
unique), i.e., there exists α0 ∈ X such that for all α ∈ X, α0 ≤ α. This
ordinal will be denoted by min X.
(O3) Whenever X is a set and f : X → ON is a function then there exists ξ ∈ ON
such that f (u) < ξ for all u ∈ X.
In future I shall not explicitly mention the above postulates but note that (O2) is
equivalent to the principle of transfinite induction on ON:
∀α (∀ξ < α ξ ∈ X → α ∈ X) → ON ⊆ X .
Definition: 4.2 Let N be the smallest set of ordinals which contains 0 and with α
also contains α+1. Then all the ordinals in N different from 0 are successor ordinals.
The first ordinal that does not belong to N is the least limit ordinal, denoted by ω.
Definition: 4.3 A class U ⊆ ON is said to be an initial segment or just a
segment if ∀α ∈ U ∀β < α β ∈ U .
Any segment is either an ordinal α (i.e. the set of ordinals < α) or the class of
ordinals ON.
Let X, Y ⊆ ON and f : X → Y be a function. For V ⊆ X let f [V ] = {f (α) |
α ∈ V }.
f is strictly increasing or order preserving if
∀α, β ∈ X (α < β → f (α) < f (β)).
f is said to be an enumeration function of Y or listing function of Y or
ordering function of Y if f is strictly increasing, f [X] = Y and X is a segment.
Given a set U ⊆ ON we denote by sup U the smallest ordinal ξ such that
∀α ∈ U α ≤ ξ.
32
Lemma: 4.4 Let X be a segment of ON and f : X → ON be order preserving.
Then α ≤ f (α) holds for all α ∈ X.
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Proof: Use transfinite induction on α.
Lemma: 4.5 Every Y ⊆ ON has a unique enumeration function EnumY .
Proof: Existence. Define the collapsing function CY : Y → ON by
CY (α) = {CY (ξ) | ξ ∈ Y ∧ ξ < α}.
Then CY is 1–1 and X := CY [Y ] is a segment. Now let
EnumY := (CY )−1 .
Here is another way of defining EnumY : Let c be a set which is not an ordinal,
e.g. c = {1}, where 1 = {0}. Define F : ON → ON by transfinite recursion via
min(Y \ {F (β) | β < α}) if Y \ {F (β) | β < α} =
6 ∅
F (α) =
c
otherwise.
Then let X := {α | F (α) ∈ Y } and EnumY (α) = F (α) for α ∈ X.
The proof that any of the above provides indeed an enumeration function for Y
is left to the reader.
Uniqueness. Let f : X → Y and g : X 0 → Y both be ordering functions of Y . Then
X ⊆ X 0 or X 0 ⊆ X since both are segments. In the first case show by induction on
α ∈ X that f (α) = g(α). But since f [X] = Y and g is 1-1 this implies X = X 0 .
The argument in the case X 0 ⊆ X is of course analogous.
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Definition: 4.6 Let X ⊆ ON. X is unbounded if for all α there exists γ ∈ X
such that γ > α.
X is closed if sup U ∈ X whenever U is a non-empty subset of X.
We use the phrase X is club or a club to convey that X is closed and unbounded.
A function f : ON → ON is continuous if f (sup U ) = sup f [U ] for all nonempty sets of ordinals U .
f : ON → ON is a normal function if f is order preserving and continuous.
Lemma: 4.7 Let Y ⊆ ON. EnumY is a normal function iff Y is closed and unbounded (Y is club).
Proof: Set f := EnumY . First suppose that f is normal. As dom(f ) = ON, Y must
be unbounded. Let V ⊆ Y be a non-empty set. Let U = f −1 [V ] = {ξ | f (ξ) ∈ V }.
Since f is continuous we have sup V = sup f [U ] = f (sup U ) ∈ Y .
Conversely assume that Y is unbounded. Then the domain of EnumY must be
ON. If Y is closed and U 6= ∅ is a set of ordinals we have sup f [U ] ∈ Y , hence
sup f [U ] = f (α) for some α. Clearly, ξ ≤ α holds for all ξ ∈ U , and hence sup U ≤ α.
On the other hand, if δ < α then f (δ) < f (ξ) for some ξ ∈ U , and hence δ < sup U .
As a result, sup U = α, thus sup f [U ] = f (sup U ).
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33
Definition: 4.8 Let ON≥α := {δ | δ ≥ α}. Define the ordinal sum α + ξ by
α + ξ := EnumON≥α (ξ).
Since ON≥α is obviously a club, the function ξ 7→ α + ξ is a normal function by
Lemma 4.7.
Lemma: 4.9 The following properties hold for ordinal addition:
1. α + 0 = α.
2. α + (ξ + 1) = (α + ξ) + 1.
3. α + λ = supξ<λ (α + ξ) for limits λ.
4. ξ < η implies α + ξ < α + η.
5. α ≤ α + ξ and ξ ≤ α + ξ.
6. α + (β + γ) = (α + β) + γ.
Proof: These are straightforward consequences of ξ 7→ α + ξ being an enumeration
function. (5) is proved by induction on γ. If γ = 0 this follows from (1). If γ = γ0 +1,
then
(2)
(2)
i.h.
(2)
α + (β + γ) = α + ((β + γ0 ) + 1) = (α + (β + γ0 )) + 1
= ((α + β) + γ0 ) + 1 = (α + β) + γ.
If γ is a limit then
i.h.
(α + β) + γ = sup((α + β) + ξ) = sup(α + (β + ξ)) ≤ α + (β + γ).
ξ<γ
ξ<γ
Suppose ζ < α + (β + γ). Then ζ < α or ζ = α + ζ0 for some ζ0 < β + γ. In
the latter case ζ0 < β or ζ0 = β + ξ for some ξ < γ. Thus in every case we have
ζ < supξ<γ (α + (β + ξ)), showing that α + (β + γ) ≤ supξ<γ (α + (β + ξ)).
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Definition: 4.10 We say that an ordinal α > 0 is an additive principal number
or additively indecomposable if ξ, η < α implies ξ + η < α.
The class of additive principal numbers we denote by AP.
Lemma: 4.11
1. Let α > 0. α ∈
/ AP iff there exist η, ξ < α such that η + ξ = α.
2. 1 is the smallest additive principal number and ω is the next one. Additive
principal number > 1 are limit ordinals.
3. Every infinite cardinal is in AP.
4. AP is a club.
34
Proof: (1) Assume α ∈
/ AP. Then α ≤ ξ + δ for some ξ, δ < α. Since α ∈ ON≥ξ
there exists η such that α = ξ + η. Hence η ≤ δ < α.
Conversely if α = ξ + η for some ξ, η < α then α ∈
/ AP.
(2) is obvious.
(3) Clearly ω ∈ AP. Let ρ be an infinite cardinal > ω. Note that if ξ, η < ρ then
the cardinalities of ξ and η are smaller than ρ and the cardinality of ξ + η is not
bigger than the maximum of the cardinalities of ξ, η, ω, and hence < ρ.
(4) To show unboundedness, take any α and define α0 = α+1 and αn+1 = αn +αn .
Let β := sup{αn | n ∈ N}. Since αn > 0 we have αn < αn + αn = αn+1 . Clearly,
α < β.
If ξ, η < β then ξ, β < αn for some n, and hence ξ + η < αn + αn = αn+1 < β.
Thus β ∈ AP.
As for closure, let U ⊆ AP be a non-empty set. Let α = sup U . If ξ, η < α then
ξ < ξ 0 and η < η 0 for some ξ 0 , η 0 ∈ U . Hence ξ + η < max(ξ 0 , η 0 ) ≤ α.
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Definition: 4.12 Let ω α := EnumAP (α).
Lemma: 4.13
1. ω 0 = 1 and ω 1 = ω.
2. ω λ = supξ<λ ω ξ .
3. If α < β then ω α < ω β .
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Proof: Obvious.
Lemma: 4.14 Let α > 0. Then α ∈ AP iff for all ξ < α, ξ + α = α.
Proof: This is true for α = 1.
Let α ∈ AP and α > 1. Then α is a limit and hence ξ + α = supδ<α (ξ + δ) ≤ α.
On the other hand, ξ + α ≥ α.
Conversely assume ξ + α = α for all ξ < α. Then if ξ, η < α we have ξ + η <
ξ + α = α, whence α ∈ AP.
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Definition: 4.15 We write α =N F α1 +. . .+αn if α = α1 +. . .+αn , α1 , . . . , αn ∈ AP
and α1 ≥ . . . ≥ αn .
Theorem: 4.16 (Cantor’s normal form, Cantor 1897) For every α > 0 there
are uniquely determined α1 , . . . , αn ∈ AP such that
α =N F α1 + . . . + αn .
Proof: We prove the existence by induction on α. If α ∈ AP, the α =N F α. If
α∈
/ AP then by Lemma 4.11 there exist 0 < η, ξ < α such that η + ξ = α. By the
inductive assumption we have η =N F η1 + . . . + ηm and ξ =N F ξ1 + . . . + ξn for some
η1 , . . . , ηm , ξ1 , . . . , ξn ∈ AP. As a result,
α =N F η1 + . . . + ηj + ξ1 + . . . + ξn
35
where j is the largest index such that ηj ≥ ξ1 . Note that there exists such a j since
η1 ≥ ξ1 for otherwise we would have η + ξ = ξ = α.
To show uniqueness assume α =N F α1 + . . . + αm and α =N F α1∗ + . . . + αn∗ .
We show m = n and αi = αi∗ by induction on m. As α1 < α1∗ would entail
α1 + . . . + αm < α1∗ we have α1 ≥ α1∗ . Thus, by symmetry, α1 = α1∗ . Hence
m = n = 1 or α2 + . . . + αm = α2∗ + . . . + αn∗ . In the latter case the induction
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hypothesis tells us that m = n and αi = αi∗ for 2 ≤ i ≤ m.
Corollary: 4.17 Let α =N F α1 + . . . + αm and β =N F β1 + . . . + βm . Then α < β
iff one of the following holds:
(i) m < n and αi = βi for all i ≤ m;
(ii) there exists j ≤ min(m, n) such that αj < βj and αi = βi holds for all 1 ≤ i <
j.
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Proof: Obvious.
Definition: 4.18 We define ordinal multiplication and exponentiation as follows:
α·0 = 0
α · (β + 1) = α · β + α
α · λ = sup{α · ξ | ξ < λ} when λ is a limit.
α0 = 1
αβ+1 = αβ · α
αλ = sup{αξ | ξ < λ} when λ is a limit.
Note that on account of Lemma 4.14, definitions 4.12 and 4.18 give rise to the same
function ξ 7→ ω ξ .
Lemma: 4.19
1. α < β and γ > 0 iff γ · α < γ · β.
2. If α ≤ β then α · γ ≤ β · γ.
3. α · (β + γ) = α · β + α · γ.
4. (α · β) · γ = α · (β · γ).
5. ω α+1 = ω α · ω.
6. ω α+β = ω α · ω β .
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Proof: Exercises.
36
4.1
Veblen’s functions
Definition: 4.20 For f : ON → ON define
Fix(f ) := {α | f (α) = α};
f 0 := EnumFix(f ) .
Veblen called f 0 the derivative of f .
Lemma: 4.21 (i) If f : ON → ON is normal then Fix(f ) is a club and f 0 is a
normal function, too.
(ii) Let ρ > 0. If Xξ is a sequence of clubs for ξ < ρ then
\
Xξ
ξ<ρ
is also a club.
Proof: (i) By Lemma 4.7 it suffices to show that Fix(f ) is a club. For unboundedness let α be arbitrary and define α0 = α + 1, αn+1 = f (αn ) and α∗ = sup{αn | n ∈
ω}. Then α∗ > α and
f (α∗ ) = sup{f (αn ) | n ∈ ω} = sup{αn+1 | n ∈ ω} = α∗
whence α∗ ∈ Fix(f ).
For closure assume U ⊆ Fix(f ) is a non-empty set. Then f (sup U ) = sup f [U ] =
sup U since f is continuous and f [U ] = U since U consists of fixed points of f . Thus
sup U ∈ Fix(f ).
(ii) Closure is obvious as each class Xξ is closed.
For unboundedness let α be arbitrary. Recursively define αn and αnξ for ξ < ρ
as follows. Set α0 = α + 1. For ξ < ρ choose αnξ in such a way that αnξ ∈ Xξ and
αnξ > αn . Let αn+1 = supξ<ρ αnξ . Put α+ = supk αk . Then we have
αn < αnξ ≤ αn+1
and hence α+ = supk αkξ for all ξ < ρ. Whence α+ ∈ Xξ for all ξ < ρ.
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Definition: 4.22 (Veblen 1908) Define
Cr(0) = AP;
Cr(α + 1) = Fix(ϕα );
\
Cr(λ) =
Cr(ξ) if λ is a limit;
ξ<λ
ϕα = EnumCr(α) .
Corollary: 4.23 For every α, Cr(α) is a club and ϕα is a normal function.
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Proof: Lemma 4.11 and Lemma 4.21.
37
Lemma: 4.24
1. If α ≤ β then Cr(β) ⊆ Cr(α).
2. ϕ0 (α) = ω α .
3. ϕα is strictly increasing.
4. β ≤ ϕα (β).
5. If α < β then Cr(β) is a proper subclass of Cr(α), ϕα (γ) ≤ ϕβ (γ), and
ϕα (ϕβ (γ)) = ϕβ (γ).
Proof: (1) follows readily by induction on β. (2) and (3) are immediate and (4)
follows from Lemma 4.4.
As to (5) note that ϕβ (γ) ∈ Cr(α + 1) by (1) and hence ϕα (ϕβ (γ)) = ϕβ (γ).
As ϕα (0) < ϕα (ϕβ (0)) = ϕβ (0) it follows that ϕα (0) ∈
/ Cr(β) and hence Cr(β) is
a proper subclass of Cr(α).
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Theorem: 4.25 (ϕ-comparison)
lowing conditions is satisfied:
(i) ϕα1 (β1 ) = ϕα2 (β2 ) holds iff one of the fol-
1. α1 < α2 and β1 = ϕα2 (β2 )
2. α1 = α2 and β1 = β2
3. α2 < α1 and ϕα1 (β1 ) = β2 .
(ii) ϕα1 (β1 ) < ϕα2 (β2 ) holds iff one of the following conditions is satisfied:
1. α1 < α2 and β1 < ϕα2 (β2 )
2. α1 = α2 and β1 < β2
3. α2 < α1 and ϕα1 (β1 ) < β2 .
Proof: We prove (i) and (ii) simultaneously.
Case 1: α1 < α2 . Then ϕα1 (ϕα2 (β2 )) = ϕα2 (β2 ) and hence
ϕα1 (β1 ) = ϕα2 (β2 )
ϕα1 (β1 ) < ϕα2 (β2 )
iff
iff
β1 = ϕα2 (β2 );
β1 < ϕα2 (β2 ).
Case 2: α1 = α2 . Then
ϕα1 (β1 ) = ϕα2 (β2 )
ϕα1 (β1 ) < ϕα2 (β2 )
iff
iff
β1 = β2 ;
β1 < β2 .
Case 3: α1 > α2 . Then ϕα2 (ϕα1 (β1 )) = ϕα1 (β1 ) and hence
ϕα1 (β1 ) = ϕα2 (β2 )
ϕα1 (β1 ) < ϕα2 (β2 )
iff
iff
β2 = ϕα1 (β1 );
ϕα1 (β1 ) < β2 .
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Corollary: 4.26 If α < β then ϕα (0) < ϕβ (0). Hence α ≤ ϕα (0).
38
Proof: The first part follows from Theorem 4.25(ii)(1). Thus the function α 7→
ϕα (0) is order preserving, so α ≤ ϕα (0) follows by Lemma 4.4.
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Theorem: 4.27 (ϕ normal form) For every α ∈ AP there exist uniquely determined ordinals ξ and η such that α = ϕξ (η) and η < α.
Proof: For existence, let ξ := min{δ | α < ϕδ (α)}. ξ exists by Corollary 4.26. If
ξ = 0 we have α = ϕ0 (η) for some η < α since α ∈ AP.
If ξ > 0 then ϕζ (α) = α for all ζ < ξ and hence α ∈ Cr(ξ) which implies
α = ϕξ (η) for some η < α.
It remains to show uniqueness. If α = ϕξ (η) = ϕξ0 (η 0 ) where η, η 0 < α, then the
cases (1) and (3) from Theorem 4.25(i) cannot hold and hence η = η 0 and ξ = ξ 0 . t
u
Definition: 4.28 Let SC := {α | ϕα (0) = α} and Γβ = EnumSC (β).
Theorem: 4.29 SC is a club and hence β 7→ Γβ is a normal function.
Proof: By Corollary 4.26 we know that α 7→ ϕα (0) is strictly increasing. One can
also show that this function is continuous. Hence its class of fixed points SC forms
a club.
t
u
Lemma: 4.30 SC = {α | α > 0 ∧ ∀ξ, η < α ϕξ (η) < α}.
t
u
Proof: Exercise.
4.2
Two ordinal representation systems
Let ε0 be the first ordinal α such that such that ω α = α. Then ∀β < ε0 β < ω β .
Another notation for ε0 is ϕ1 (0).
Also note that if ρ ∈ AP and ρ < Γ0 then there exist (unique) α, β < ρ such that
α = ϕα (β).
Definition: 4.31
(i) The set OT(ε0 ) is inductively defined by the following clauses:
1. 0 ∈ OT(ε0 ).
2. If α1 , . . . , αn ∈ OT(ε0 ) ∩ AP and α1 ≥ . . . ≥ αn and n > 1 then α1 + . . . +
αn ∈ OT(ε0 ).
3. If α ∈ OT(ε0 ) then ω α ∈ OT(ε0 ).
(ii) OT(Γ0 ) is inductively defined by the following clauses:
1. 0 ∈ OT(Γ0 ).
2. If α1 , . . . , αn ∈ OT(Γ0 ) ∩ AP and α1 ≥ . . . ≥ αn and n > 1 then α1 +
. . . + αn ∈ OT(Γ0 ).
3. If α, β ∈ OT(Γ0 ) and α, β < ϕα (β) then ϕα (β) ∈ OT(Γ0 ).
Corollary: 4.32
(i) OT(ε0 ) = ε0 .
39
(ii) OT(Γ0 ) = Γ0 .
Proof: Use induction on α < ε0 to show that α ∈ OT(ε0 ). Similarly, use induction
on α < Γ0 to show that α ∈ OT(Γ0 ).
t
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Ordinals β < Γ0 have a unique normal form, namely either β = 0 or β =N F
β1 + . . . βn with β1 , βn ∈ AP and n > 1 or β =N F ϕγ (δ) with γ, δ < β. Thus every
0 < β < Γ0 can be uniquely represented in terms of smaller ordinals which again can
be uniquely represented in terms of yet smaller ordinals and 0 etc. As this process
terminates after finitely many steps, every β < Γ0 has a unique term representation
over the alphabet 0, +, ϕ.
Corollary: 4.33 There is a primitive recursive set A0 ⊆ N, a primitive recursive
relation ≺ on A0 and primitive binary recursive functions +̂ and ϕ̂ such that
f : (OT(Γ0 ), <, +, ϕ) ∼
= (A0 , ≺, +̂, ϕ̂)
for some structural isomorphism f . Moreover
(OT(ε0 ), <, +, ϕ0 ) ∼
= (B0 , ≺1 , +̂, ϕ̂0 ),
where B0 = {x ∈ A0 | x ≺ f (ε0 )}, ≺1 is the restriction of ≺ to A0 and +̂ and ϕ̂0
are the restrictions of these functions to B0 .
Proof: Ordinals < Γ0 can be coded by natural numbers. For instance a coding
function
d . e : Γ0 −→ N
could be defined as follows:

if α = 0
 0
h1, dα1 e, . . . , dαn ei if α =N F α1 + · · · + αn where n > 1
dαe =

h2, dα1 e, dα2 ei
if α =N F ϕα1 (α2 )
where hk1 , · · · , kn i := 2k1 +1 · . . . · pknn +1 with pi being the ith prime number (or any
other coding of tuples). Further define:
A0 := range of d.e
dαe +̂ dβe := dα + βe
dαe ≺ dβe :⇔ α < β
ϕ̂(dαe, dβe) := dϕα (β)e.
Then
hΓ0 , +, ϕ, <i ∼
= hA0 , +̂, ϕ̂, ≺i.
It remains to show that A0 , ≺, +̂, ϕ̂ are primitive recursive. This can be seen by
defining them via a simultaneous primitive recursive definition, viewing Corollary
4.18 and Theorem 4.25 as the recursive clauses for defining ≺.
t
u
40
5
Ordinal analysis of PA and some subsystems of
second order arithmetic
The most important structure in mathematics is arguably the structure of the natural numbers N = (N; 0N , 1N , +N , ×N , E N , <N ), where 0N denotes zero, 1N denotes
the number one, +N , ×N , E N denote the successor, addition, multiplication, and exponentiation function, respectively, and <N stands for the less-than relation on the
natural numbers. In particular, E N (n, m) = nm .
Many of the famous theorems and problems of mathematics such as Fermat’s
and Goldbach’s conjecture, the Twin Prime conjecture, and Riemann’s hypothesis
can be formalized as sentences of the language of N and thus concern questions
about the structure N.
Definition: 5.1 A theory designed with the intent of axiomatizing the structure N
is Peano arithmetic, PA. The language of PA has the predicate symbols =, <,
the function symbols +, ×, E (for addition, multiplication,exponentiation) and the
constant symbols 0 and 1. The Axioms of PA comprise the usual equations and laws
for addition, multiplication, exponentiation, and the less-than relation. In addition,
PA has the Induction Scheme
(IND)
A(0) ∧ ∀x[A(x) → A(x + 1)] → ∀xA(x)
for all formulae A(a) of the language of PA.
Gentzen showed that transfinite induction up to the ordinal
ω
ε0 = sup{ω, ω ω , ω ω , . . .} = least α. ω α = α
suffices to prove the consistency of PA. To appreciate Gentzen’s result it is pivotal to
note that he applied transfinite induction up to ε0 solely to elementary computable
predicates and besides that his proof used only finitistically justified means. Hence,
a more precise rendering of Gentzen’s result is
F + EC-TI(ε0 ) ` Con(PA),
(40)
where F signifies a theory that embodies only finitistically acceptable means, EC-TI(ε0 )
stands for transfinite induction up to ε0 for elementary computable predicates, and
Con(PA) expresses the consistency of PA. Finally, we should spell out the scheme
EC-TI(ε0 ) in the language of PA:
∀x [∀y (y ≺ x → P (y)) → P (x)] → ∀x P (x)
for all elementary computable predicates P .
Gentzen also showed that his result was the best possible in that PA proves transfinite induction up to α for arithmetic predicates for any α < ε0 . The compelling
picture conjured up by the above is that the non-finitist part of PA is encapsulated
in EC-TI(ε0 ) and therefore “measured” by ε0 , thereby tempting one to adopt the
following definition of proof-theoretic ordinal of a theory T :
|T |Con = least α. F + EC-TI(α) ` Con(T ).
41
(41)
In the above, many notions were left unexplained. We will now consider them one
by one. The elementary computable functions are exactly the Kalmar elementary
functions, i.e. the class of functions which contains the successor, projection, zero,
addition, multiplication, and modified subtraction functions and is closed under
composition and bounded sums and products. A predicate is elementary computable
if its characteristic function is elementary computable.
According to an influential analysis of finitism due to W.W. Tait, finististic
reasoning coincides with a system known as primitive recursive arithmetic. For the
purposes of ordinal analysis, however, it suffices to identify F with an even more
restricted theory known as Elementary Recursive Arithmetic, EA. EA is a weak
subsystem of PA having the same defining axioms for +, ×, E, < but with induction
restricted to elementary computable predicates.
We shall add a new unary predicate symbol U to the language of PA which will
serve the purpose of a free predicate variable.
Definition: 5.2 We shall formalize PAU in the sequent calculus. In addition to
the rules of the (classical) sequent calculus we have to add the following axioms:
(=ref) Γ ⇒ ∆, t = t
(=sym) Γ, s = t ⇒ ∆, t = s
(=tran) Γ, s1 = s2 , s2 = s3 ⇒ ∆, s1 = s3
(=sub) Γ, s1 = t1 , . . . , sn = tn , A(s1 , . . . , sn ) ⇒ ∆, A(t1 , . . . , tn )
only for atomic formulas A(~s ).
(suc1) Γ ⇒ ∆, suc(s) 6= 0 and Γ, suc(s) = suc(t) ⇒ ∆, s = t.
(+) Γ ⇒ ∆, s + 0 = 0 and Γ ⇒ ∆, s + suc(t) = suc(s + t).
(·) Γ ⇒ ∆, s · 0 = 0 and Γ ⇒ ∆, s · suc(t) = s · t + s.
(IND) Γ, A(0), ∀x [A(x) → A(x + 1)] ⇒ ∆, ∀xA(x)
for all formulas A(a).
As the ultimate goal of this course is to carry out an ordinal analysis of a system
of set theory, we shall not particularly dwell on an ordinal analysis of PA. To
soften the ascend to set theory, however, we will first give an ordinal analysis of two
subsystems of second order arithmetic. The analysis of PAU will arise as a corollary.
Ordinal analysis is concerned with theories serving as frameworks for formalising
significant parts of mathematics. It is known that virtually all of ordinary mathematics can be formalized in Zermelo-Fraenkel set theory with the axiom of choice,
ZFC. Hilbert and Bernays [25] showed that large chunks of mathematics can already be formalized in second order arithmetic. Owing to these observations, proof
theory has been focusing on set theories and subsystems of second order arithmetic.
Further scrutiny revealed that a small fragment is sufficient. Under the rubric of
Reverse Mathematics a research programme has been initiated by Harvey Friedman
some thirty years ago. The idea is to ask whether, given a theorem, one can prove
42
its equivalence to some axiomatic system, with the aim of determining what prooftheoretical resources are necessary for the theorems of mathematics. More precisely,
the objective of reverse mathematics is to investigate the role of set existence axioms
in ordinary mathematics. The main question can be stated as follows:
Given a specific theorem τ of ordinary mathematics, which set existence
axioms are needed in order to prove τ ?
Central to the above is the reference to what is called ‘ordinary mathematics’. This
concept, of course, doesn’t have a precise definition. Roughly speaking, by ordinary
mathematics we mean main-stream, non-set-theoretic mathematics, i.e. the core
areas of mathematics which make no essential use of the concepts and methods
of set theory and do not essentially depend on the theory of uncountable cardinal
numbers.
Subsystems of second order arithmetic. The framework chosen for studying
set existence in reverse mathematics, though, is second order arithmetic rather than
set theory. Second order arithmetic, Z2 , is a two-sorted formal system with free
and bound first order variables (also called numerical variables; the same as for PA)
and free set variables U0 , U1 , U2 , . . . as well as bound set variables X0 , X1 , X2 , . . .
supposed to range over sets of natural numbers. The language L2 of second-order
arithmetic also contains the symbols of PA, and in addition has a binary relation
symbol ∈ for elementhood. Formulae are built from atomic formulae s = t, s < t,
and s ∈ U (where s, t are numerical terms, i.e. terms of PA) by closing off under
the connectives ∧, ∨, →, ¬, numerical quantifiers ∀x, ∃x, and set quantifiers ∀X, ∃X.
The basic arithmetical axioms in all theories of second-order arithmetic are the
defining axioms for 0, 1, +, ×, E, < (as for PA) and the induction axiom
∀X(0 ∈ X ∧ ∀x(x ∈ X → x + 1 ∈ X) → ∀x(x ∈ X)).
We consider the axiom schema of C-comprehension for formula classes C which is
given by
C − CA
∃X∀u(u ∈ X ↔ F (u))
for all formulae F ∈ C in which X does not occur. Natural formula classes are the
arithmetical formulae, consisting of all formulae without second order quantifiers
∀X and ∃X, and the Π1n -formulae, where a Π1n -formula is a formula of the form
∀X1 . . . QXn A(X1 , . . . , Xn ) with ∀X1 . . . QXn being a string of n alternating set
quantifiers, commencing with a universal one, followed by an arithmetical formula
A(X1 , . . . , Xn ).
ACA0 denotes the theory consisting of the basic arithmetical axioms plus the
scheme
∃X∀u(u ∈ X ↔ F (u))
for all arithmetical formula F (a) in which X does not occur. ACA denotes the
theory ACA0 augmented by the scheme of induction for all L2 -formulae.
43
5.1
The semi-formal system RA∗ of Ramified Analysis
Definition: 5.3 RA∗ has the following symbols:
• Bound number variables: x0 , x1 , x2 , . . ..
• Free predicate variables of level α for each ordinal α < Γ0 : U0α , U1α , U2α . . .
• Bound predicate variables of level β for each ordinal 0 < β < Γ0 : X0β , X1β , X2β , . . ..
• The symbols 0, suc, +, ·.
• The logical symbols ∧, ∨, →, ¬, ∀, ∃ and λ.
• Symbols for primitive recursive functions and relations.
• Parentheses.
Inductive definition of formulas and predicators.
1. Every numerical atomic formula is a formula of level 0.
2. Every free predicate variable of level α is a predicator of level α.
3. If P α is a predicator of level α and t is a term, then t ∈ P α is a formula of
level α.
4. If A and B are formulas of level α and β then A ∧ B, A ∨ B, A → B are
formulas of level max(α, β) and ¬A is a formula of level α.
5. If F (0) is a formula of level α and x is a bound number variable which does not
occur in F (0), then ∀x F (x) and ∃x F (x) are formulae of level α and λx F (x)
is a predicator of level α.
6. If U β is a free predicate variable of level β 6= 0, F (U β ) is a formula of level α
and X β a bound predicate variable of level β which does not occur in F , then
∀X β F (X β ) and ∃X β F (X β ) are formulae of level max(α, β).
Inductive definition of the length |A| of a formula A.
1. Every atomic numerical formula A has length 0, |A| = 0.
2. |U α (t)| = ω · α.
3. If A and B are formulas then |A ∧ B| = |A ∨ B| = |A → B| = max(|A|, |B|) + 1
and |¬A| = |A| + 1.
4. |∀x F (x)| = |∃x F (x)| = |λx F (x)| = |F (0)| + 1.
5. |∀X β F (X β )| = |∃X β F (X β )| = max(ω · β, |F (U 0 )| + 1).
Definition: 5.4 We define the infinitary proof system RA∗ . A true (false) atomic
formula is an atomic formula without free variables (and hence closed) which is true
(false) on the standard interpretation. The axioms of ACA∞ are the following:
44
(A1) Γ ⇒ ∆, A where A is a true atomic formula.
(A2) Γ, A ⇒ ∆ where A is a false atomic formula.
(A3) Γ, U α (s) ⇒ ∆, U α (t) where s and t have the same numerical value and U α is
a free set variable.
The inference rules of RA∗ comprise those of the sequent calculus with the exception
of (∀R) and (∃L). The latter are replaced by two infinitary rules, i.e. rules with
infinitely many premisses. They correspond to the so called ω-rule:
Γ ⇒ ∆, F (0); Γ ⇒ ∆, F (1); . . . ; Γ ⇒ ∆, F (n); . . .
Γ ⇒ ∆, ∀x F (x)
F (0), Γ ⇒ ∆; F (1), Γ ⇒ ∆; . . . ; F (n), Γ ⇒ ∆; . . .
∃x F (x), Γ ⇒ ∆
(ωR)
(ωL)
The price to pay will be that deductions become infinite objects, i.e. infinite wellfounded trees.
We will also need rules for the higher order quantifiers and predicators. Variables
P, P0 , P1 , . . . will range over predicators and variables P α , P0α , P1α , . . . will range over
predicators of level α. We write lev(P ) for the level of P . Pβ stands the collection
of predicators with levels < β.
F (t), Γ ⇒ ∆
PL
λxF (x)(t), Γ ⇒ ∆
F (P ), Γ ⇒ ∆,
Γ ⇒ ∆, F (P ) all P ∈ Pβ
∀β L
∀X β F (X β ), Γ ⇒ ∆
F (P ), Γ ⇒ ∆ all P ∈ Pβ
β
Γ ⇒ ∆, F (t)
PR
Γ ⇒ ∆, λxF (x)(t)
β
Γ ⇒ ∆, ∀X β F (X β )
Γ ⇒ ∆, F (P )
∃β L
β
∃X F (X ), Γ ⇒ ∆
β
∀β R
∃β R
Γ ⇒ ∆, ∃X F (X )
where in ∀β L and ∃β R, P is a predicator of level < β.
Definition: 5.5 RA∗
α
ρ
Γ ⇒ ∆ is defined inductively as follows:
(i) If Γ ⇒ ∆ is an axiom, then RA∗
α
ρ
Γ ⇒ ∆ for any α, ρ.
αi
(ii) If RA∗ ρ Γi ⇒ ∆i holds for all premisses Γi ⇒ ∆i of an inference of RA∗
other than (Cut) with conclusion Γ ⇒ ∆ and αi < α holds for all i, then
α
RA∗ ρ Γ ⇒ ∆ .
α1
(iii) If RA∗ ρ Γ, C ⇒ ∆ , RA∗
α
RA∗ ρ Γ ⇒ ∆ .
Lemma: 5.6
α1
ρ
Γ ⇒ ∆, C , |C| < ρ and α1 , α2 < α, then
(i) If B is a formula of level α then |A| = ω · α + n for some n < ω.
45
(ii) For every formula A(U ) and comprehension term P α with α < β,
|A(P α )| < |∀X β A(X β )|, |∃X β A(X β )|.
t
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Proof: Exercise.
Lemma: 5.7 For every formula C of RA∗ ,
RA∗
2·|A|
0
Γ, C ⇒ ∆, C .
Proof: Use induction on |A|.
t
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We list some technical lemmata that will be useful for proving cut elimination.
Lemma: 5.8 (Substitution) Let Γ(s) and ∆(s) be sets of formulas with some
occurrences of s indicated and let t be a term with the same numerical value.
If
α
ρ
Γ(s) ⇒ ∆(s) , then
α
ρ
Γ(t) ⇒ ∆(t) .
t
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Proof: Use induction on α.
Lemma: 5.9 (Weakening)
α
If RA∗ ρ Γ ⇒ ∆ , Γ ⊆ Γ0 and ∆ ⊆ ∆0 , then RA∗
α
ρ
Γ0 ⇒ ∆0 .
t
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Proof: Use induction on α.
Lemma: 5.10 (Inversion)
(i) If RA∗
α
ρ
Γ, A ∧ B ⇒ ∆ then RA∗
α
ρ
Γ, A, B ⇒ ∆ .
(ii) If RA∗
α
ρ
Γ ⇒ ∆, A ∧ B then RA∗
α
ρ
Γ ⇒ ∆, A and RA∗
α
ρ
Γ ⇒ ∆, B .
(iii) If RA∗
α
ρ
Γ, A ∨ B ⇒ ∆ then RA∗
α
ρ
Γ, A ⇒ ∆ and RA∗
α
ρ
Γ, B ⇒ ∆ .
(iv) If RA∗
α
ρ
Γ ⇒ ∆, A ∨ B then RA∗
α
ρ
Γ ⇒ ∆, A, B .
(v) If RA∗
α
ρ
Γ ⇒ A → B, ∆ then RA∗
α
ρ
A, Γ ⇒ ∆, B .
(vi) If RA∗
α
ρ
Γ, A → B ⇒ ∆ then RA∗
α
ρ
Γ ⇒ ∆, A and RA∗
(vii) If RA∗
α
ρ
Γ ⇒ ¬A, ∆ then RA∗
α
ρ
Γ, A ⇒ ∆ .
(viii) If RA∗
α
ρ
Γ, ¬A ⇒ ∆ then RA∗
α
ρ
Γ ⇒ ∆, A .
(ix) If RA∗
α
ρ
Γ ⇒ ∆, ∀x B(x) then RA∗
α
ρ
Γ ⇒ ∆, B(s) for any closed term s.
(x) If RA∗
α
ρ
Γ, ∃x B(x) ⇒ ∆ then RA∗
α
ρ
Γ, B(s) ⇒ ∆ for any closed term s.
α
ρ
Γ, B ⇒ ∆ .
α
α
ρ
Γ ⇒ ∆, B(P ) for any predicator
α
α
ρ
Γ, B(P ) ⇒ ∆ for any predicator
(xi) If RA∗ ρ Γ ⇒ ∆, ∀X β B(X β ) then RA∗
P ∈ Pβ .
(xii) If RA∗ ρ Γ, ∃X β B(X β ) ⇒ ∆ then RA∗
in P ∈ Pβ .
46
(xiii) If RA∗
α
ρ
Γ ⇒ ∆, λxF (x)(t) then RA∗
α
ρ
Γ ⇒ ∆, F (t) .
(xiv) If RA∗
α
ρ
Γ, λxF (x)(t) ⇒ ∆ then RA∗
α
ρ
Γ, F (t) ⇒ ∆ .
t
u
Proof: All are provable by easy inductions on α.
Lemma: 5.11 (Reduction)
α
Suppose ρ ≤ |C|. If RA∗ ρ Γ, C ⇒ ∆ and RA∗
RA∗
α#α#β#β
|C|
β
ρ
Ξ ⇒ Θ, C , then
Γ, Ξ ⇒ ∆, Θ .
Proof: The proof is by induction on α#α#β#β and very similar to Lemma 2.17.
We only look at two cases where C and was the principal formula of the last inference
in both derivations.
Case 1: The first is when C is of the form ∀X β A(X β ). Then we have
RA∗
α1
ρ
Γ, C, A(P 0 ) ⇒ ∆
RA∗
βP
ρ
Ξ ⇒ Θ, C, A(P )
and
for some α1 < α and predicator P 0 ∈ Pβ as well as βP < β for all predicators
P ∈ Pβ . By the induction hypothesis we obtain
α1 #α1 #β#β
RA∗
and
RA∗
|C|
α#α#βP 0 #βP 0
|C|
α#α#β#β
Cutting out A(P 0 ) gives RA∗
|C|
Γ, Ξ, A(P 0 ) ⇒ ∆, Θ
Γ, Ξ ⇒ ∆, Θ, A(P 0 ) .
Γ, Ξ ⇒ ∆, Θ .
Case 2: The second case is when C is of the form ∀x A(x) Then we have
RA∗
α1
ρ
Γ, C, A(t) ⇒ ∆
RA∗
βn
ρ
Ξ ⇒ Θ, C, A(n)
and
for some α1 < α and closed term t as well as βn < β for all numbers n. Let m be
the numerical value of t. By Lemma 5.10(ix) we have
RA∗
α1
ρ
Γ, C, A(m) ⇒ ∆ .
By the induction hypothesis we thus get
α1 #α1 #β#β
RA∗
and
RA∗
|C|
α#α#βm #βm
|C|
Cutting out A(m) gives RA∗
α#α#β#β
|C|
Γ, Ξ, A(m) ⇒ ∆, Θ
Γ, Ξ ⇒ ∆, Θ, A(m) .
Γ, Ξ ⇒ ∆, Θ .
47
t
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Theorem: 5.12 (First Cut Elimination Theorem)
α
4α
If RA∗ δ+1 Γ ⇒ ∆ then RA∗ δ Γ ⇒ ∆ .
Proof: We use induction on α. If Γ ⇒ ∆ is an axiom then we clearly get the
desired result. So let’s assume that Γ ⇒ ∆ is not an axiom. Then we have a last
inference (I) with premisses Γi ⇒ ∆i . Suppose the inference was not a cut or a
αi
cut of a degree < δ. We then have RA∗ δ Γi ⇒ ∆i for some αi < α. By the
induction hypothesis we have RA∗
4α
RA∗ δ
4αi
δ
α
Γi ⇒ ∆i . Applying the same inference (I)
yields
Γ ⇒ ∆ since 4αi < 4 .
Now suppose the last inference was a cut with a cut formula C satisfying |C| = δ.
By the induction hypothesis we have
RA∗
4α1
RA∗
4α2
and
δ
δ
Γ, C ⇒ ∆
Γ ⇒ ∆, C
for some α1 , α2 < n. We can then apply the Reduction Lemma 5.11 to these
4α1 #4α1 #4α2 #4α2
derivations and arrive at RA∗ δ
Γ ⇒ ∆ . Since 4α1 #4α1 #4α2 #4α2 ≤
4α the desired conclusion follows.
t
u
Theorem: 5.13 (Second Cut Elimination Theorem)
If RA∗
α
ρ+ω ν
Γ ⇒ ∆ then RA∗
ϕν (α)
ρ
Γ ⇒ ∆.
Proof: We use induction on ν with a subsidiary induction on α. The assertion
holds for ν = 0 by the First Cut Elimination Theorem 5.12. Now suppose ν > 0.
If Γ ⇒ ∆ is an axiom then we clearly get the desired result. So let’s assume that Γ ⇒ ∆ is not an axiom. Then we have a last inference (I) with
premisses Γi ⇒ ∆i . Suppose the inference was not a cut or a cut of rank < ρ.
αi
We then have RA∗ ρ+ων Γi ⇒ ∆i for some αi < α. By the subsidiary induction
hypothesis we have RA∗
ϕν (αi )
ρ
Γi ⇒ ∆i . Applying the same inference (I) yields
ϕν (α)
RA∗ ρ
Γ ⇒ ∆.
Now suppose the last inference was a cut with cut formula C such that ρ ≤
|C| < ρ + ω ν . Then there exist ν0 < ν and n < ω such that |C| < ρ + ω ν0 · n. After
performing a cut with C we have
RA∗
ϕν (α)
ρ+ω ν0 ·n
Γ ⇒ ∆.
We also have ϕν0 (ϕν (α)) = ϕν (α). Therefore by n-fold application of the main
induction hypothesis we obtain RA∗
5.2
ϕν (α)
ρ
Γ ⇒ ∆.
t
u
Interpretation of subsystems of Z2 in RA∗
To facilitate the interpretation of subsystems of Z2 in RA∗ we will assume that they
are formalized via the sequent calculus.
48
Definition: 5.14 The sequent calculus version of ACA0 has all the axioms of PAU
given in Definition 5.2 but with IND excluded. Further axioms are:
(IA) Γ ⇒ ∆, ∀X [0 ∈ X ∧ ∀u (u ∈ X → u + 1 ∈ X) → ∀u u ∈ X].
(A-CA) Γ ⇒ ∆, ∃Y ∀u [u ∈ Y ↔ A(u)]
where A(a) is an arithmetic formula in which Y does not occur.
In addition to the usual inference rules of the sequent calculus we also need inference
rules for the second order quantifiers:
F (V ), Γ ⇒ ∆,
∀L
∀X F (X), Γ ⇒ ∆
Γ ⇒ ∆, F (U )
∀R
Γ ⇒ ∆, ∀X F (X)
F (U ), Γ ⇒ ∆
∃L
∃X F (X), Γ ⇒ ∆
Γ ⇒ ∆, F (V )
∃R
Γ ⇒ ∆, ∃X F (X)
where the variable U in ∀2 R and ∃2 L is an eigenvariable of the respective inference,
i.e. U is not to occur in the lower sequent.
The sequent calculus version of ACA also has the axiom scheme (IND) from Definition 5.2.
The theory of ∆11 -analysis (that’s the name Schütte gave it in [52, VIII.20]) or
1
(∆1 − CR) comprises ACA and in addition has the rule of ∆11 -comprehension:
⇒ ∀x [∀XA(X, x) ↔ ∃Y B(Y, x)]
Γ ⇒ ∆, ∃Z∀x [x ∈ Z ↔ ∀XA(X, x)]
∆11 -CR
where A(U, a) and B(U, a) are arithmetic formulae. Note that the premiss of an
instance of ∆11 -CR does not have any side formulas.
Definition: 5.15 Let 0 < σ < Γ0 . Let Ξ ⇒ Θ be an L2 -sequent. We call an
L∗RS -sequent Ξσ ⇒ Θσ a σ-instance of Ξ ⇒ Θ if is obtained by the following
steps:
1. Write Ξ ⇒ Θ as
Ξ(a1 , . . . , ak , U1 , . . . , Ur ) ⇒ Θ(a1 , . . . , ak , U1 , . . . , Ur )
fully indicating all free variables occurring in it.
2. Replace every free variable ai by a number mi and every variable Uj by a
predicator Pj of level < σ.
3. Finally add to every bound variable occurring in
Ξ(m1 , . . . , mk , P1 , . . . , Pr ) ⇒ Θ(m1 , . . . , mk , P1 , . . . , Pr )
a superscript σ (i.e., X changes to X σ ) and the result is Ξσ ⇒ Θσ .
If Γ ⇒ ∆ is a sequent of PAU , we say that Γ0 ⇒ ∆0 is a numerical instance of
Γ ⇒ ∆ if it is obtained by the following steps:
49
1. Write Γ ⇒ ∆ as Γ(a1 , . . . , an ) ⇒ ∆(a1 , . . . , an ), where all free number variables are fully indicated.
2. Replace every ai by the same numeral mi .
3. In Γ(m1 , . . . , mn ) ⇒ ∆(m1 , . . . , mn ) replace every expression U (t) by t ∈ U00 ,
and the result is Γ0 ⇒ ∆0 .
Lemma: 5.16 RA∗
2·|F (0)|+ω
F (0), ∀x [F (x) → F (x + 1)] ⇒ ∀x F (x)
0
Proof: We show
2·(|F (0)|+n)
RA∗
0
F (0), ∀x [F (x) → F (x + 1)] ⇒ F (n)
(42)
by induction on n. Let η := |F (0)|. By Lemma 5.7 we have
RA∗
2·η
RA∗
2·(η+n)
0
F (0), ∀x [F (x) → F (x + 1)] ⇒ F (0) .
Assume
0
F (0), ∀x [F (x) → F (x + 1)] ⇒ F (n) .
(43)
2·η
We have RA∗ 0 F (n + 1) ⇒ F (n + 1) by Lemma 5.7 and thus via an inference
(→ L) we obtain
RA∗
2·(η+n)+1
0
F (0), ∀x [F (x) → F (x + 1)], F (n) → F (n + 1) ⇒ F (n + 1) .
Using (∀L) we arrive at
RA∗
2·(η+n)+2
0
F (0), ∀x [F (x) → F (x + 1)] ⇒ F (n + 1)
(44)
which is what we want as 2 · (η + n) + 2 = 2 · (η + n + 1).
As a consequence of (42) we get the desired assertion via an inference (ωR). t
u
Theorem: 5.17 (First Interpretation Theorem)
there exist n, k < ω such that
RA∗
ω+n
k
(i) If PAU
Γ ⇒ ∆ then
Γ0 ⇒ ∆0
holds for every numerical instance of Γ0 ⇒ ∆0 of Γ ⇒ ∆.
(ii) If ACA0
that
∀X A(X) where A(U ) is arithmetic then there exist n, k < ω such
RA∗
(iii) If ACA
ω+n
k
∀X 1 A(X 1 ) .
Γ ⇒ ∆ then there exist n, k < ω such that
RA∗
ω+ω+n
ω+k
Γ1 ⇒ ∆1
holds for every 1-instance Γ1 ⇒ ∆1 of Γ ⇒ ∆.
50
Proof: (i) Use induction on the length of the derivation in PAU . Numerical instances of the axioms of PAU other than (IA) are axioms of RA∗ . (IA) is deducible
cut free and with length ω + 1 by Lemma 5.16. For the induction step note that
inferences of PAU other than (∀R) and (∃L) are inferences of RA∗ too. If the last
inference was (∀R) use (ωR) instead and if it was (∃L) use (ωL). Also note that if
A is numerical instance of a formula of PAU then |A| < ω.
(ii) follows from (i) since if ACA0 ∀X A(X) with A(U ) is arithmetic then
PAU
A(U ) .
(iii) Again use induction on the length of the derivation. Note that a 1-instance
of a formula of ACA has length < ω + ω.
t
u
Theorem: 5.18 (Second Interpretation Theorem) If (∆11 -CR)
RA∗
ω·σ+ω+6·n
ω·σ+ω
n
Γ ⇒ ∆ then
Γσ ⇒ ∆σ
holds for any σ = ω n · β with β > 0 and σ-instance Γσ ⇒ ∆σ of Γ ⇒ ∆.
t
u
Proof: Homework #5 Problem 5.
Corollary: 5.19
(i) If PAU
Γ ⇒ ∆ then there exists α < ε0 such that
α
RA∗
0
Γ0 ⇒ ∆0
holds for every numerical instance of Γ0 ⇒ ∆0 of Γ ⇒ ∆.
(ii) If ACA0 ∀XA(X) where A(U ) is arithmetic and has no free number variables then there exists α < ε0 such that
RA∗
(iii) If ACA
α
0
∀X 1 A(X 1 ) .
Γ ⇒ ∆ then there exists α < εε0 such that
α
RA∗
0
Γ1 ⇒ ∆1
holds for every 1-instance Γ1 ⇒ ∆1 of Γ ⇒ ∆.
(iv) If (∆11 -CR) ∀XA(X) where A(U ) is arithmetic and has no free number
variables then there exists α < ϕω (0) such that
RA∗
α
0
∀X 1 A(X 1 ) .
51
6
The limits of the deducibility of transfinite induction
Definition: 6.1 Let ≺ be a relation on N. For a formula F (a) define Define
Prog(≺, F ) := ∀x (∀y ≺ x F (y) → F (x));
TI(≺, F ) := Prog(≺, F ) → ∀x F (x).
Also define
Prog(≺, U ) := ∀x (∀y ≺ x y ∈ U → x ∈ U );
TI(≺, U ) := Prog(≺, U ) → ∀x x ∈ U.
If ≺ is well-founded we define
|n|≺ = sup{|k|≺ + 1 | k ≺ n}
k≺k = sup{|n|≺ | n ∈ N}
For a theory T whose language comprises that of PAU define
k T ksup = sup{k≺k| T ` TI(≺, U ) where ≺ is primitive recursive}.
Definition: 6.2 We define the notion of a U -positive (U -negative) formula of PAU .
A formula in which U does not occur is both U -positive and U -negative. A formula
t ∈ U is U -positive and ¬t ∈ U is U -negative. If A, B and F (a) are U -positive
(U -negative) then so are A ∧ B, A ∨ B, ∀xF (x) and ∃xF (x). If A is U -positive
(U -negative) then ¬A is U -negative (U -positive). If A is U -negative (U -positive)
and B is U -positive (U -negative) then A → B is U -positive (U -negative).
If A(U ) is a formula of PAU without free number variables and X ⊆ N we write
(N, X) |= A(U )
if A(U ) becomes true on interpreting U by X.
Note that if A(U ) is U -positive, X ⊆ Y ⊆ N and (N, X) |= A(U ), then (N, Y ) |=
A(U ). We shall refer to this fact as monotonicity of U -positive formulae. Similarly,
U -negative formulae behave in an anti-monotonic way.
W
V
If Γ is a non-empty finite set of formulae we denote by Γ and V Γ the disjunction and conjunction
of all formula in Γ, respectively. Also define ∅ to be the
W
formula 0 = 0 and ∅ to be the formula 0 = 1.
Proposition: 6.3 Assume that ≺ is a well-founded relation on N which is defined
by an arithmetic formula, i.e. there is an arithmetic formula B(a, b) with exactly
the exhibited free variables such that n ≺ m iff B(n, m) holds in the standard model.
Let ∆ be a finite set of U -positive arithmetic formulae and Γ be a finite set of
U -negative arithmetic formulae with no other free variables than U . We identify U
with U00 .
If δ = max(|t1 |≺ , . . . , |tr |≺ ) and
RA∗
β
0
t1 ∈ U, . . . , tr ∈ U, Prog(≺, U ), Γ ⇒ ∆
then
(N, {m | |m|≺ < δ + 2β }) |=
52
^
Γ→
_
∆.
Proof: We employ induction on β. If the entire sequent is an axiom one readily
checks that the claim is true. If the last inference introduced a principal formula
belonging to Γ or ∆ the claim follows readily from the induction hypothesis applied
to the premisses. Now assume that the last inference had Prog(≺, U ) as its principal
formula. Then we have
RA∗
β0
0
t1 ∈ U, . . . , tr ∈ U, Prog(≺, U ), ∀y ≺ t y ∈ U → t ∈ U, Γ ⇒ ∆
for some closed term t and β0 < β. Using (→ L)-inversion we get
RA∗
β0
t
0 1
β0
RA∗ 0 t1
∈ U, . . . , tr ∈ U, Prog(≺, U ), Γ ⇒ ∆, ∀y ≺ t y ∈ U ;
(45)
∈ U, . . . , tr ∈ U, t ∈ U, Prog(≺, U ), Γ ⇒ ∆ .
(46)
Note that ∀y ≺ t y ∈ U is a U -positive formula, and hence we may apply the
induction hypothesis to (45) to arrive at
^
_
Γ → ( ∆ ∨ ∀y ≺ t y ∈ U ).
(N, {m | |m|≺ < δ + 2β0 }) |=
V
W
If (N, {m | |m|≺ < δ + 2β0 }) |= Γ → ∆ we are done owing to monotonicity. If
the latter is not the case, then we have
(N, {m | |m|≺ < δ + 2β0 }) |= ∀y ≺ t y ∈ U
which entails that |t|≺ ≤ δ + 2β0 . As a result, the induction hypothesis applied to
(46) with δ 0 = δ + 2β0 yields
^
_
(N, {m | |m|≺ < δ 0 + 2β0 }) |=
Γ→
∆.
As δ 0 + 2β0 = δ + 2β0 + 2β0 < δ + 2β we are done again by monotonicity.
t
u
Corollary: 6.4 If
β
RA∗
0
Prog(≺, U ) → ∀x x ∈ U
then k≺k ≤ 2β .
Proof: The assumption entails that
RA∗
β
0
Prog(≺, U ) ⇒ ∀x x ∈ U ,
and hence by the previous Proposition, |n|≺ < 2β holds for all n, whence k≺k ≤ 2β .
t
u
Corollary: 6.5
(i) k PAU ksup = ε0 .
(ii) k ACA0 ksup = ε0 .
(iii) k ACA ksup = εε0 .
(iv) k (∆11 -CR) ksup = ϕω (0).
Proof: The “≤” estimates follow from Corollary 5.19 in combination with Corollary 6.4. The “≥” estimates in (i),(ii),(iii) follow from homework assignment #6,
problems 3 and 4. The “≥” part in (iv) will be another exercise.
t
u
53
6.1
Proof-theoretical reductions
Ordinal analyses of theories allow one to compare the strength of theories. This subsection defines the notions of proof-theoretic reducibility and proof-theoretic strength
that will be used henceforth.
All theories T considered in the following are assumed to contain a modicum
of arithmetic. For definiteness let this mean that the system PRA of Primitive
Recursive Arithmetic is contained in T , either directly or by translation.
Definition: 6.6 Let T1 , T2 be a pair of theories with languages L1 and L2 , respectively, and let Φ be a (primitive recursive) collection of formulae common to both
languages. Furthermore, Φ should contain the closed equations of the language of
PRA.
We then say that T1 is proof-theoretically Φ-reducible to T2 , written T1 ≤Φ T2 , if
there exists a primitive recursive function f such that
PRA ` ∀φ ∈ Φ ∀x [ProofT1 (x, φ) → ProofT2 (f (x), φ)].
(47)
T1 and T2 are said to be proof-theoretically Φ-equivalent, written T1 ≡Φ T2 , if T1 ≤Φ
T2 and T2 ≤Φ T1 .
The appropriate class Φ is revealed in the process of reduction itself, so that in
the statement of theorems we simply say that T1 is proof-theoretically reducible to T2
(written T1 ≤ T2 ) and T1 and T2 are proof-theoretically equivalent (written T1 ≡ T2 ),
respectively. Alternatively, we shall say that T1 and T2 have the same proof-theoretic
strength when T1 ≡ T2 .
Feferman’s notion of proof-theoretic reducibility (in S. Feferman: Hilbert’s program relativized: Proof-theoretical and foundational reductions, J. Symbolic Logic
53 (1988) 364–384) is more relaxed in that he allows the reduction to be given by a
T2 -recursive function f , i.e.
T2 ` ∀φ ∈ Φ ∀x [ProofT1 (x, φ) → ProofT2 (f (x), φ)].
(48)
The disadvantage of (48) is that one forfeits the transitivity of the relation ≤Φ .
Furthermore, in practice, proof-theoretic reductions always come with a primitive
recursive reduction, so nothing seems to be lost by using the stronger notion of
reducibility.
6.2
The general form of ordinal analysis
In this subsection I attempt to say something general about all ordinal analyses that
have been carried out thus far. One has to bear in mind that these concern “natural”
theories. Also, to circumvent countless and rather boring counter examples, I will
only address theories that have at least the strength of PA and and always assume
the pertinent ordinal representation systems are closed under α 7→ ω α .
Before delineating the general form of an ordinal analysis, we need several definitions. We first garner some features (following that ordinal representation systems
used in proof theory always have, and collectively call them “elementary ordinal
representation system”. One reason for singling out this notion is that it leads to
an elegant characterization of the provably recursive functions of theories equipped
with transfinite induction principles for such ordinal representation systems.
54
Definition: 6.7 Elementary recursive arithmetic, EA, is a weak system of number
theory, in a language with 0, 1, +, ×, E (exponentiation), <, whose axioms are:
1. the usual recursion axioms for +, ×, E, <.
2. induction on ∆0 -formulae with free variables.
EA is referred to as elementary recursive arithmetic since its provably recursive
functions are exactly the Kalmar elementary functions, i.e. the class of functions
which contains the successor, projection, zero, addition, multiplication, and modified subtraction functions and is closed under composition and bounded sums and
products
Definition: 6.8 For a set X and and a binary relation ≺ on X, let LO(X, ≺)
abbreviate that ≺ linearly orders the elements of X and that for all u, v, whenever
u ≺ v, then u, v∈X.
A linear ordering is a pair hX, ≺i satisfying LO(X, ≺).
Definition: 6.9 An elementary ordinal representation system (EORS) for a limit
ordinal λ is a structure hA, , n 7→ λn , +, ×, x 7→ ω x i such that:
(i) A is an elementary subset of N.
(ii) is an elementary well-ordering of A.
(iii) || = λ.
(iv) Provably
in EA, λn is a proper initial segment of for each n, and
S
λ
n = . In particular, EA ` ∀y λy ∈ A ∧ ∀x ∈ A∃y [x λy ].
n
(v) EA ` LO(A, )
(vi) +, × are binary and x 7→ ω x is unary. They are elementary functions on
elementary initial segments of A. They correspond to ordinal addition, multiplication and exponentiation to base ω, respectively. The initial segments of
A on which they are defined are maximal.
n 7→ λn is an elementary function.
(vii) hA, , +, ×, ω x i satisfies “all the usual algebraic properties” of an initial segment of ordinals. In addition, these properties of hA, , +, ×, ω x i can be
proved in EA.
(viii) Let ñ denote the nth element in the ordering of A. Then the correspondence
n ↔ ñ is elementary.
(ix) Let α = ω β1 + · · · + ω βk , β1 ≥ · · · ≥ βk (Cantor normal form). Then the
correspondence α ↔ hβ1 , . . . , βk i is elementary.
Elements of A will often be referred to as ordinals, and denoted α, β, . . ..
55
Definition: 6.10 Suppose LO(A, ) and F (u) is a formula. Then TIhA,i (F ) is the
formula
∀n ∈ A [∀x nF (x) → F (n)] → ∀n ∈ A F (n).
(49)
TI(A, ) is the schema consisting of TIhA,i (F ) for all F .
Given a linear ordering hA, i and α ∈ A let Aα = {β ∈ A : β α} and α be the
restriction of to Aα .
In what follows, quantifiers and variables are supposed to range over the natural
numbers. When n denotes a natural number, n̄ is the canonical name in the language
under consideration which denotes that number.
Observation: 6.11 Every ordinal analysis of a classical or intuitionistic theory T
that has ever appeared in the literatureSprovides an EORS hA, , . . .i such that T is
proof-theoretically reducible to PA + α∈A TI(Aᾱ , ᾱ ).
S
Moreover, if T is a classical theory, then T and PA + α∈A TI(Aᾱ , ᾱ ) prove
the sameSarithmetic sentences, whereas if T is based on intuitionististic, then T and
HA + α∈A TI(Aᾱ , ᾱ ) prove the same arithmetic sentences.
Furthermore, k T ksup =k k.
Remark: 6.12 There is a lot of leeway in stating the latter observation. For instance, instead of PA one could take PRA or EA as the base theory,
Sand the scheme
0
of transfiniteSinduction could be restricted to Σ1 formulae as PA + α∈A TI(Aᾱ , ᾱ )
and EA + α∈A Σ01 -TI(Aᾱ , ᾱ ) have the same proof-theoretic strength, providing
that A is closed under exponentiation α 7→ ω α .
Observation 6.11 lends itself to a formal definition of the notion of proof-theoretic
ordinal of a theory T . Of course, before one can go about determining the prooftheoretic ordinal of T , one needs to be furnished with representations of ordinals.
Not surprisingly, a great deal of ordinally informative proof theory has been concerned with developing and comparing particular ordinal representation systems.
Assuming that a sufficiently strong EORS hA, , . . .i has been provided, we define
[
|T |hA,,...i := least ρ ∈ A. T ≡ PA +
TI(Aᾱ , ᾱ )
(50)
αρ
and call |T |hA,,...i , providing this ordinal exists, the proof-theoretic ordinal of T with
respect to hA, , . . .i.
Since, in practice, the ordinal representation systems used in proof theory are
comparable, we shall frequently drop mentioning of hA, , . . .i and just write |T | for
|T |hA,,...i .
Note, however, that |T |hA,,...i might not exist even if the order-type of is
bigger than k T ksup . A simple example is provided by the theory PA + Con(PA)
(where Con(PA) expresses the consistency of PA) when we take hA, , . . .i to be a
standard EORS for ordinals > ε0 ; the reason
S being that PA + Con(PA) is prooftheoreticallySstrictly stronger than PA + αε0 TI(Aᾱ , ᾱ ) but also strictly weaker
than PA + αε0 +1 TI(Aᾱ , ᾱ ). Therefore, as opposed to k · ksup , the norm |·|hA,,...i
is only partially defined and does not induce a prewellordering on theories T with
k T ksup <k k.
The remainder of this subsection expounds on important consequences of ordinal
analyses that follow from Observation 6.11.
56
S
S
Proposition: 6.13 PA + α∈A TI(Aᾱ , ᾱ ) and HA + α∈A TI(Aᾱ , ᾱ ) prove the
same sentences in the negative fragment, where a sentence is in the negative fragment
if it is built from atomic formulae via ∧, →, ¬, ∀x.
S
S
Proof: PA + α∈A TI(Aᾱ , ᾱ ) can be interpreted in HA + α∈A TI(Aᾱ , ᾱ ) via
the Gödel–Gentzen ¬¬-translation. Observe that for an instance of the schema of
transfinite induction we have
(∀u [∀x (∀y [y ≺ x → φ(y)] → φ(x)) → φ(u)])¬¬ ≡
(∀u [∀x (∀y [¬¬y ≺ x → ¬¬φ(y)] → ¬¬φ(x)) → ¬¬φ(u)]).
Thus for primitive recursive ≺ the ¬¬-translation is HA equivalent to an instance
of the same schema.
t
u
Corollary: 6.14 PA +
same Π01 sentences.
S
α∈A
TI(Aᾱ , ᾱ ) and HA +
S
α∈A
TI(Aᾱ , ᾱ ) prove the
Since many well-known and important theorems as well as conjectures from number
theory are expressible in Π01 form (examples: the quadratic reciprocity law, Wiles’
theorem, also known as Fermat’s conjecture, Goldbach’s conjecture, the Riemann
hypothesis), Π01 conservativity ensures that many mathematically important theorems which turn out to be provable in S will be provable in T , too.
However, Π01 conservativity is not always a satisfactory conservation result. Some
important number-theoretic statements are Π02 (examples are: the twin prime conjecture, miniaturized versions of Kruskal’s theorem, totality of the van der Waerden
function), and in particular, formulas that express the convergence of a recursive
function for all arguments. Consider a formula ∀n ∃m P (n, m), where P (n, m) is
a primitive recursive formula expressing that “m codes a complete computation of
algorithm A on input n.” The ¬¬-translation of this formula is ∀n ¬∀m ¬P (n, m),
conveying the convergence of the algorithm A for all inputs only in a weak sense.
Fortunately, Proposition 6.14 can be improved to hold for sentences of Π02 form.
S
S
Proposition: 6.15 PA + α∈A TI(Aᾱ , ᾱ ) and HA + α∈A TI(Aᾱ , ᾱ ) prove the
same Π02 sentences.
The missing link to get from Proposition 6.13 to Proposition 6.15 is usually
provided by Markov’s Rule for primitive recursive predicates, MRP R : if ¬∀n¬Q(n)
(or, equivalently, ¬¬ ∃n Q(n)) is a theorem, where Q is a primitive recursive relation,
then ∃n Q(n) is a theorem. Kreisel [30] showed that MRP R holds for HA. A
variety of intuitionistic systems have since been shown to be closed under MRP R ,
using a variety of complicated methods, notably Gödel’s dialectica interpretation and
normalizability. A particularly elegant and short proof for closure under MRP R is
due to Friedman [18] and, independently, to Dragalin [12]. However, though the
Friedman–Dragalin argument
works for a host of systems, it doesn’t seem to work
S
in the case of HA + α∈A TI(Aᾱ , ᾱ ).
Proof of Proposition 6.15: We will give a direct proof, i.e. without using Proposition 6.13. So suppose
[
PA +
TI(Aᾱ , ᾱ ) ` ∀x ∃y φ(x, y),
α∈A
57
where φ is ∆0 . Then there already exists a δ ∈ A such that
PA + TI(Aδ̄ , δ̄ ) ` ∀x ∃y φ(x, y).
(51)
We now use the coding of infinitary PA∞ derivations presented in [53], section 4.2.2.
β
Let d ρ ψ signify that d is the code of a PA∞ derivation with length ≤ β, cut-rank
ρ and end formula ψ. (51) implies that there is a d0 and n < ω such that
[
δ·ω
HA +
TI(Aᾱ , ᾱ ) ` d0 n ∀x ∃y φ(x, y) .
(52)
α∈A
To obtain a cut-free proof of ∀x ∃y φ(x, y) in PA∞ one needs transfinite induction
γ
γ
:= ω ωm . This
up to the ordinal ωnδ·ω , where ω0γ := γ and ωm+1
S amount of transfinite
induction is available in our background theory HA + α∈A TI(Aᾱ , ᾱ ) as A is
closed under ξ 7→ ω ξ . Also note that the cut-elimination procedure is completely
effective. Thus from (52) we obtain, for some d∗ ,
HA +
[
TI(Aᾱ , ᾱ ) ` d∗
δ·ω
ωn
0
∀x ∃y φ(x, y) ,
(53)
α∈A
and further
HA +
[
TI(Aᾱ , ᾱ ) ` ∀x∃d d
δ·ω
ωn
0
∃y φ(ẋ, y)
(54)
α∈A
(where Feferman’s dot convention has been used here). Let TrΣ1 be a truth predicate
for Gödel numbers of disjunctions of Σ1 formulae (cf. [59], section 1.5, in particular
1.5.7). We claim that
[
_
β
HA +
TI(Aᾱ , ᾱ ) ` ∀d ∀β ≤ ωnδ·ω ∀Γ ⊆ Σ1 [ d 0 Γ → TrΣ1 ( Γ)],
(55)
α∈A
where ∀Γ ⊆ W
Σ1 is a quantifier ranging over Gödel numbers of finite sets of Σ1
formulae and Γ stands for the Gödel number corresponding to the disjunction of
all formulae of Γ. (55) is proved by induction on β by observing that all formulae
occurring in a cut-free PA∞ proof of a set of Σ1 formulae are Σ1 themselves and
the only inferences therein are either axioms or instances of the (∃) rule or improper
instances of the ω rule. Combining (54) and (55) we obtain
[
HA +
TI(Aᾱ , ᾱ ) ` ∀x TrΣ1 ( ∃y φ(ẋ, y) ).
(56)
α∈A
As
HA ` ∀x [ TrΣ1 ( ∃y φ(ẋ, y) ) ↔ ∃y φ(x, y)]
(cf. [59], Theorem 1.5.6), we finally obtain
[
HA +
TI(Aᾱ , ᾱ ) ` ∀x ∃y φ(x, y).
α∈A
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58
In section 2 we considered the ordinal |T |Con . What is the relation between
|T |Con and |T |hA,,...i ? First we have to delineate the meaning of |T |Con , though.
The latter is only determined with respect to a given ordinal representation system
hB, ≺, . . .i. Thus let
|T |Con = least α ∈ B. PRA + PR-TI(α) ` Con(T ).
It turns out that S
the two ordinals are the same when T is proof-theoretically reducible to PA + α∈A TI(Aᾱ , ᾱ ), A is closed under α 7→ ω α and hB, ≺, . . .i is a
proper end extension of hA, , . . .i. The reasons are as follows:
S
Proposition: 6.16 The consistency of PA + α∈A TI(Aᾱ , ᾱ ) can be proved in
the theory PRA + PR-TI(A, ), where PR-TI(A, ) stands for transfinite induction
along for primitive recursive predicates.
Hint of proof. First note that PRA + PR-TI(A, ) ` Π01 -TI(A, ). The key to
showing this is that for each α ∈ A and each x ∈ ω we can code α and x by the
ordinal ω · α + x which is less than ω · (α + 1) and therefore inSA.
Secondly, one has to show that an ordinal analysis of PA + α∈A TI(Aᾱ , ᾱ ) can
0
be carried
S out in PRA + Π1 -TI(A, ). The main tool to achieve this is to embed
PA+ α∈A TI(Aᾱ , ᾱ ) into a system of Peano arithmetic with an infinitary rule, the
so-called ω-rule, and a repetition rule, Rep, which simply repeats the premise as the
conclusion. The ω-rule allows one to infer ∀xφ(x) from the infinitely many premises
φ(0̄), φ(1̄), φ(2̄), . . . (where n̄ denotes the nth numeral); its addition accounts for
the fact that the infinitary system enjoys cut-elimination. The addition of the
Rep rule enables one to carry out a continuous cut elimination, due to Mints [35],
which is a continuous operation in the usual tree topology on prooftrees. A further
pivotal step consists in making the ω-rule more constructive by assigning codes to
proofs, where codes for applications of finitary rules contain codes for the proofs
of the premises, and codes for applications of the ω-rule contain Gödel numbers
for primitive recursive functions enumerating codes of the premises. Details can
be found in [53]. The main idea here is that we can do everything with primitive
recursive proof–trees instead of arbitrary derivations. A proof–tree is a tree, with
each node labelled by: A sequent, a rule of inference or the designation “Axiom”,
two sets of formulas specifying the set of principal and minor formulas,respectively,
of that inference, and two ordinals (length and cut–rank) such that the sequent is
obtained from those immediately above it through application of the specified rule
of inference. The well-foundedness of a proof–tree is then witnessed by the (first)
ordinal “tags” which are in reverse order of the tree order. As a result, the notion
of being a (code of a) proof tree is Π01 . The cut elimination for infinitary proofs
with finite cut rank (as presented in [53]) can be formalized in PRA + Π01 -TI(A, ).
The last step consists in recognizing that every endformula of Π01 form of a cut free
infinitary proof is true. The latter employs Π01 -TI(A, ). For details see [53].
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59
7
Kripke-Platek Set Theory
One of the fragments of ZF which has been studied intensively is Kripke-Platek
set theory, KP. Its standard models are called admissible sets. One of the reasons
that this is a truly remarkable theory is that a great deal of set theory requires only
the axioms of KP. An even more important reason is that admissible sets have
been a major source of interaction between model theory, recursion theory and set
theory. (cf. [4]1 ). KP arises from ZF by completely omitting the Powerset axiom
and restricting Separation and Collection to absolute predicates (cf. [4]), i.e. ∆0
formulas. These alterations are suggested by the informal notion of ‘predicative’.
The axiom systems for set theories considered in this paper are formulated in
the usual language of set theory (called L∈ hereafter) containing ∈ as the only nonlogical symbol besides =. Formulae are built from prime formulae a ∈ b and a = b
by use of propositional connectives and quantifiers ∀x, ∃x. Quantifiers of the forms
∀x ∈ a, ∃x ∈ a are called bounded. Bounded or ∆0 -formulae are the formulae wherein
all quantifiers are bounded; Σ1 -formulae are those of the form ∃xϕ(x) where ϕ(a) is
a ∆0 -formula. For n > 0, Πn -formulae (Σn -formulae) are the formulae with a prefix
of n alternating unbounded quantifiers starting with a universal (existential) one
followed by a ∆0 -formula. The class of Σ-formulae is the smallest class of formulae
containing the ∆0 -formulae which is closed under ∧, ∨, bounded quantification and
unbounded existential quantification.
One of the set theories which is amenable to ordinal analysis is Kripke-Platek
set theory, KP. Its standard models are called admissible sets. One of the reasons
that this is an important theory is that a great deal of set theory requires only
the axioms of KP. An even more important reason is that admissible sets have
been a major source of interaction between model theory, recursion theory and set
theory (cf. [4]). KP arises from ZF by completely omitting the power set axiom
and restricting separation and collection to bounded formulae. These alterations
are suggested by the informal notion of ‘predicative’.
Definition: 7.1 By a ∆0 formula or bounded formula we mean a formula of set
theory in which all the quantifiers appear restricted, that is have one of the forms
(∀x ∈ b) or (∃x ∈ b).
The axioms of KP are:
Extensionality:
∀x (x ∈ a ↔ x ∈ b) → a = b.
Set Induction:
∀x[∀y ∈ x G(y) → G(x)] → ∀xG(x)
Pair:
Union:
∃x (x = {a, b}).
S
∃x (x = a).
Infinity:
∃x [x 6= ∅ ∧ (∀y ∈ x)(∃z ∈ x)(y ∈ z)].
∆0 Separation:
∃x ∀u[u ∈ x ↔ (u ∈ a ∧ F (u))]
for all ∆0 –formulas F
∆0 Collection:
(∀x ∈ a)∃yG(x, y) → ∃z(∀x ∈ a)(∃y ∈ z)G(x, y)
for all ∆0 –formulas G.
1
J. Barwise: Admissible sets and structures. (Springer, Berlin, 1975)
60
To be more precise, the axioms of KP consist of Extensionality, Pair, Union,
Infinity, Bounded Separation
∃x ∀u [u ∈ x ↔ (u ∈ a ∧ F (u))]
for all bounded formulae F (u), Bounded Collection
∀x ∈ a ∃y G(x, y) → ∃z ∀x ∈ a ∃y ∈ z G(x, y)
for all bounded formulae G(x, y), and Set Induction
∀x [(∀y ∈ x H(y)) → H(x)] , → ∀x H(x)
for all formulae H(x).
A transitive set A such that (A, ∈) is a model of KP is called an admissible
set. Of particular interest are the models of KP formed by segments of Gödel’s
constructible hierarchy L. The constructible hierarchy is obtained by iterating the
definable powerset operation through the ordinals
L0 = ∅,
[
Lλ =
{Lβ : β < λ} λ limit
Lβ+1 = {X : X ⊆ Lβ ; X definable over hLβ , ∈ i}.
So any element of L of level α is definable from elements of L with levels < α and
the parameter Lα . An ordinal α is admissible if the structure (Lα , ∈) is a model of
KP.
Formulae of L2 can be easily translated into the language of set theory. Some of
the subtheories of Z2 considered above have set-theoretic counterparts, characterized
by extensions of KP. KPi is an extension of KP via the axiom
∀x∃y[x∈y ∧ y is an admissible set].
(Lim)
KPl denotes the system KPi without Bounded Collection. It turns out that
(Π11 −AC) + BI proves the same L2 -formulae as KPi, while (Π11 −CA) proves the
same L2 -formulae as KPl.
The intuitionistic version of KP, will be denoted by IKP.
By IKP0 we denote the system IKP bereft of Set Induction.
7.1
Basic principles
The intent of this section is to explore which of the well known provable consequences
of KP carry over to IKP.
7.1.1
Ordered Pairs
By the Pairing axiom, for sets a, b we get a set y such that
∀x(x ∈ y ↔ x = a ∨ x = b).
61
This set is unique by Extensionality; we call this set {a, b}. {a} = {a, a} is the set
whose unique element is a. ha, bi = {{a}, {a, b}} is the ordered pair of a and b. We
claim that if ha, bi = hc, di then a = c and b = d.
The usual classical proof argues by cases depending, for example, whether or
not a = b. This method is not available here as we cannot assume that instance of
the classical law of excluded middle. Instead we can argue as follows. Assume that
ha, bi = hc, di.
As {a} is an element of the left hand side it is also an element of the right hand
side and so either {a} = {c} or {a} = {c, d}. In either case a = c.
As {a, b} is an element of the left hand side it is also an element of the right
hand side and so either {a, b} = {c} or {a, b} = {c, d}. In either case b = c or b = d.
If b = c then a = c = b so that the two sets in ha, bi are equal and hence {c} = {c, d}
giving c = d and hence b = d. So in either case b = d.
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We will also have use for ordered triples ha, b, ci, ordered quadruples ha, b, c, di,
etc. They are defined by iterating the ordered pairs formation as follows: hai = a
and ha1 , . . . , ar , ar+1 i = hha1 , . . . , ar i, ar+1 i.
Proposition: 7.2 (IKP0 ) If c, d are sets then so is the class c × d.
Proof: Let c, d be sets. Then, as
{a} × d = {ha, bi | b ∈ d}
is a set, by Replacement, so is
c×d=
[
({a} × d)
a∈c
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by Replacement and Union.
Definition: 7.3 The collection of Σ formulae is the smallest collection containing
the ∆0 formulae closed under conjunction, disjunction, bounded quantification and
unbounded existential quantification. The collection of Π formulae is the smallest collection containing the ∆0 formulae closed under conjunction, disjunction,
bounded quantification and unbounded universal quantification.
Given a formula A and a variable w not appearing in A, we write Aw for the
result of replacing each unbounded quantifier ∃x and ∀x in A by ∃x ∈ w and ∀x ∈ w,
respectively.
Lemma: 7.4 For each Σ formula the following are intuitionistically valid:
(i) Au ∧ u ⊆ v → Av ,
(ii) Au → A.
Proof: Both facts are proved by induction following the inductive definition of Σ
formula.
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62
Theorem: 7.5 (Σ Reflection Principle). For all Σ formulae A we have the
following:
IKP0 ` A ↔ ∃aAa .
(Here a is any set variable not occurring in A; we will not continue to make these annoying conditions on variables explicit.) In particular, every Σ formula is equivalent
to a Σ1 formula in IKP0 .
Proof: We know from the previous lemma that ∃a Aa → A, so the axioms of IKP0
come in only in showing A → ∃a Aa . proof is by induction on A, the case for ∆0
formulae being trivial. We take the three most interesting cases, leaving the other
two to the reader.
Case 0. If A is ∆0 then A ↔ Aa holds for every set a.
Case 1. A is B ∧ C. By induction hypothesis, IKP0 ` B ↔ ∃a B a and IKP0 `
C ↔ ∃a C a . Let us work in IKP0 , assuming B ∧ C. Now there are a1 , a2 such that
B a1 , C a2 , so let a = a1 ∪ a2 . Then B a and C a hold by the previous lemma, and
hence Aa .
Case 2. A is B ∨ C. By induction hypothesis, IKP0 ` B ↔ ∃a B a and IKP0 `
C ↔ ∃a C a . Let us work in IKP0 , assuming B ∨ C. Then B a1 for some set a1 or
there is a set a2 such that C a2 . In the first case we have B a ∨ C a with a := a1 while
in the second case we have B a ∨ C a with a := a2 .
Case 2. A is ∀u ∈ v B(u). The inductive assumption yields IKP0 ` B(u) ↔
∃a B(u)a . Again, working in IKP0 , assume ∀u ∈ v B(u) and show ∃a ∀u ∈ v B(u)a .
b
For each u ∈ v there is a b such that
S B(u) , so by ∆0 Collection there is an a0 such
b
that ∀u ∈ v ∃b ∈ a0 B(u) . Let a = a0 . Now, for every u ∈ v, we have ∃b ⊆ a B(u)b ;
so ∀u ∈ vB(u)a , by the previous lemma.
Case 3. A is ∃u B(u). Inductively we have IKP0 ` B(u) ↔ ∃b B(u)b . Working
in IKP0 , assume ∃u B(u). Pick u0 such B(u0 ) and b such that B(u0 )b . Letting a =
b ∪ {u0 } we get u0 ∈ a and B(u0 )a by the previous lemma. Thence ∃a ∃u ∈ a B(u)a .
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In Platek’s original definition of admissible set he took the Σ Reflection Principle
as basic. It is very powerful, as we’ll see below. ∆0 Collection is easier to verify,
however.
Theorem: 7.6 (The Strong Σ Collection Principle). For every Σ formula A the
following is a theorem of IKP0 : If ∀x ∈ a ∃yA(x, y) then there is a set b such that
∀x ∈ a ∃y ∈ b A(x, y) and ∀y ∈ b ∃x ∈ a A(x, y).
Proof: Assume that
∀x ∈ a∃y ∈ b A(x, y).
By Σ Reflection there is a set c such that
∀x ∈ a ∃y ∈ c A(x, y)c .
(57)
b = {y ∈ c| ∃x ∈ a A(x, y)c },
(58)
Let
by ∆0 Separation. Now, since A(x, y)c → A(x, y) by 7.4, (57) gives us ∀x ∈ a ∃y ∈ b A(x, y),
whereas (58) gives us ∀y ∈ b ∃x ∈ a A(x, y).
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63
Theorem: 7.7 (Σ Replacement). For each Σ formula A(x, y) the following is a
theorem of IKP0 : If ∀x ∈ a ∃!y A(x, y) then there is a function f , with dom(f ) = a,
such that ∀x ∈ a A(x, f (x)).
Proof: By Σ Reflection there is a set d such that
∀x ∈ a ∃y ∈ d A(x, y)d .
Since A(x, y)d implies A(x, y) we get ∀x ∈ a ∃!y ∈ d A(x, y)d . Thus, defining f =
{hx, yi ∈ a × d| A(x, y)d } by ∆0 Separation, f is a function satisfying dom(f ) = a
and ∀x ∈ a A(x, f (x)).
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The above is sometimes infeasible because of the uniqueness requirement ∃! in the
hypothesis. In these situations it is usually the next result which comes to the
rescue.
Theorem: 7.8 (Strong Σ Replacement). For each Σ formula A(x, y) the following
is a theorem of IKP0 : If ∀x ∈ a ∃y A(x, y) then there is a function f with dom(f ) =
a such that for all x ∈ a, f (x) is inhabited and ∀x ∈ a ∀y ∈ f (x) A(x, y).
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Proof: Exercise.
One principle of KP that is not provable in IKP is ∆1 Separation.
Proposition: 7.9 (KP0 ) (∆1 Separation). If A is a Σ formula A and B is a Π
formula, then
KP0 ` ∀x ∈ a [A(x) ↔ B(x)] → ∃z ∀u[u ∈ z ↔ (u ∈ a ∧ A(x))].
Proof: The reason is that classically ∀x ∈ a [A(x) ↔ B(x)] entails ∀x ∈ a [A(x) ∨
¬B(x)] which is classically equivalent to a Σ formula.
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7.2
Σ Recursion in IKP
The mathematical power of KP resides in the possibility of defining Σ functions by
∈-recursion and the fact that many interesting functions in set theory are definable
by Σ Recursion. Moreover the scheme of ∆0 Separation allows for an extension with
provable Σ functions occurring in otherwise bounded formulae.
Proposition: 7.10 (Definition by Σ Recursion in IKP.) If G is a total (n + 2)–ary
Σ definable class function of IKP, i.e.
IKP ` ∀~xyz∃!u G(~x, y, z) = u
then there is a total (n + 1)–ary Σ class function F of IKP such that2
IKP ` ∀~xy[F (~x, y) = G(~x, y, (F (~x, z)|z ∈ y))].
2
(F (~x, z)|z ∈ y) := {hz, F (~x, z)i : z ∈ y}
64
Proof: Let A(f, ~x) be the formula
[f is a function] ∧ [dom(f ) is transitive] ∧ [∀y ∈ dom(f ) (f (y) = G(~x, y, f |y))].
Set
B(~x, y, f ) = [A(f, ~x) ∧ y ∈ dom(f )].
Claim
IKP ` ∀~x, y∃!f B(~x, y, f ).
Proof of Claim: By ∈ induction on y. Suppose ∀u ∈ y ∃g B(~x, u, g). By Strong Σ Collection we S
find a set A such that ∀u ∈ y ∃g ∈ A B(~x, u, g) and ∀g ∈ A∃u ∈ y B(~x, u, g).
Let f0 = {g : g ∈ A}. By our general assumption there exists a u0 such that
G(~x, y, (f0 (u)|u ∈ y)) = u0 . Set f = f0 ∪ {hy, u0 i}. Since for all g ∈ A, dom(g) is
transitive we have that dom(f0 ) is transitive. If u ∈ y, then u ∈ dom(f0 ). Thus
dom(f ) is transitive and y ∈ dom(f ). We have to show that f is a function. But
it is readily shown that if g0 , g1 ∈ A, then ∀x ∈ dom(g0 ) ∩ dom(g1 )[g0 (x) = g1 (x)].
Therefore f is a function. This also shows that ∀w ∈ dom(f )[f (w) = G(~x, w, f |w)],
confirming the claim (using Set Induction).
Now define F by
F (~x, y) = w := ∃f [B(~x, y, f ) ∧ f (y) = w].
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Corollary: 7.11 There is a Σ function TC of IKP such that
[
IKP ` ∀a[TC(a) = a ∪ {TC(x) : x ∈ a}].
Proposition: 7.12 (Definition by TC–Recursion) Under the assumptions of Proposition 7.10 there is an (n + 1)–ary Σ class function F of IKP such that
IKP ` ∀~xy[F (~x, y) = G(~x, y, (F (~x, z)|z ∈ TC(y)))].
Proof: Hint: Let C(f, ~x, y) be the Σ formula
[f is a function] ∧ [dom(f ) = TC(y)] ∧ [∀u ∈ dom(f )[f (u) = G(~x, u, f |TC(u))]].
Prove by ∈–induction that ∀y∃!f C(f, ~x, y).
65
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8
An Ordinal representation system for the BachmannHoward ordinal
Serving as a miniature example of an ordinal analysis of an impredicative system,
we carry out an ordinal analysis of KP. The first step is to find a sufficiently strong
ordinal representation system.
Definition: 8.1 Let Ω be a “big” ordinal, e.g. Ω = ℵ1 . By recursion on α we define
sets B(α) and the ordinal ψΩ (α) as follows:

 closure of {0, Ω} under:
+, (ξ 7→ ω ξ ), (ξ, η 7→ ϕξ (η),
B(α) =
(59)

(ξ 7−→ ψΩ (ξ))ξ<α
ψΩ (α)
=
min{ρ < Ω | ρ ∈
/ B(α) }
(60)
if the set {ρ < Ω | ρ ∈
/ B(α) } is non-empty.
As per definition ψΩ α might not be defined but the next Lemma shows that it is a
total function.
Lemma: 8.2
(i) B(α) is a countable set.
(ii) ψΩ (α) is always defined and ψΩ (α) < Ω.
S
Proof: (i) B(α) = n<ω Bn (α) where B0 (α) = {0, Ω} and
Bn+1 (α) = Bn (α) ∪ {η + δ | η, δ ∈ Bn (α) ∪ {ϕη (δ) | η, δ ∈ Bn (α)}
∪ {ψΩ (ξ) | ξ ∈ Bn (α) ∧ ξ < α}.
Inductively, each of the sets Bn (α) is countable (actually finite) and therefore B(α)
is countable.
(ii) Ω is assumed to be a regular uncountable cardinal, thus B(α) ∩ Ω cannot be
unbounded in Ω.
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Lemma: 8.3
(i) If α ≤ δ then B(α) ⊆ B(δ) and ψΩ (α) ≤ ψΩ (δ).
(ii) If α ∈ B(δ) ∩ δ then ψΩ (α) < ψΩ (δ).
(iii) If α ≤ δ and [α, δ) ∩ B(α) = ∅ then B(α) = B(δ).
S
(iv) If λ is a limit then B(λ) = ξ<λ B(ξ).
Proof: (i): B(α) ⊆ B(δ) is clearly true if α ≤ δ. And thus ψΩ (α) ≤ ψΩ (δ) follows
by definition and Lemma 8.2.
(ii): From α ∈ B(δ) ∩ δ we get ψΩ (α) ∈ B(δ) and also, by (i), ψΩ (α) ≤ ψΩ (δ).
Since ψΩ (δ) ∈
/ B(δ) this entails ψΩ (α) < ψΩ (δ).
(iii): By induction on n one easily shows that Bn (δ) ⊆ B(α). This is obvious for
n = 0. Assume it is true for n. If β < δ and β ∈ Bn (δ) then inductively we have
β ∈ B(α) and hence β < α, yielding ψΩ (β) ∈ B(α). Thus we get Bn+1 (δ) ⊆ B(α).
66
S
(iv): By (i) we S
have ξ<λ B(ξ) ⊆ B(λ). To show the reverse inclusion we only
need to show that ξ<λ B(ξ) is closed under theSoperations that define B(λ). This
is obvious for + and ϕ. So assume that δ ∈ ξ<λ B(ξ) ∩ λ. Then δ < ξ0 and
δ ∈ B(ξ1S
) for some ξ0 , ξ1 < λ. Thus, letting ξ ∗ = max(ξ0 , ξ1 ), we have ψΩ (δ) ∈
∗
B(ξ ) ⊆ ξ<λ B(ξ).
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Lemma: 8.4 ψΩ (α) ∈ SC, i.e., ϕψΩ (α) (0) = ψΩ (α).
Proof: If ψΩ (α) = ξ + δ for some ξ, δ < ψΩ (α), then ξ, η ∈ B(α) and therefore
ψΩ (α) = ξ + δ ∈ B(α), contradicting the definition of ψΩ (α).
Likewise, if ψΩ (α) = ϕρ (η) for some ρ η < ψΩ (α) then ρ, η ∈ B(α) and therefore
ψΩ (α) = ϕρ (η) ∈ B(α), contradicting the definition of ψΩ (α). Thus ψΩ (α) ∈ SC
follows by Lemma 4.30.
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Theorem: 8.5 B(α) ∩ Ω = ψΩ (α).
Proof: Clearly, ψΩ (α) ⊆ B(α) ∩ Ω.
To conclude equality, it suffices to show that X := ψΩ (α) ∪ {δ ∈ B(α) | δ ≥ Ω}
is closed under the operations that define B(α). closure of X under + and ϕ follows
from Lemma 8.4. To show closure under ψΩ for arguments < α, assume β ∈ X and
β < α. Then ψΩ (β) < ψΩ (α) by Lemma 8.3(ii), and hence ψΩ (β) ∈ X.
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Corollary: 8.6 If λ is a limit then ψΩ (λ) = supξ<λ ψΩ (ξ).
Proof:
[
ψΩ (λ) = B(λ) ∩ Ω = (
B(ξ)) ∩ Ω
ξ<λ
=
[
(B(ξ) ∩ Ω) =
[
ψΩ (ξ) = sup ψΩ (ξ).
ξ<λ
ξ<λ
ξ<λ
Here the first and fourth equality follow from Theorem 8.5 while the second equality
is a consequence of Lemma 8.3(iv).
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Γ
Definition: 8.7 Let β denote the least ordinal ρ > β such that ρ ∈ SC.
Lemma: 8.8
Γ
(i) ψΩ (α + 1) ≤ (ψΩ (α)) .
Γ
(ii) α ∈ B(α + 1) implies ψΩ (α + 1) = (ψΩ (α)) .
(iii) α ∈
/ B(α) implies B(α) = B(α + 1) and ψΩ (α + 1) = ψΩ (α).
Proof: (i): It suffices to show that
Γ
Y := (B(α + 1) ∩ (ψΩ (α)) ) ∪ {δ ∈ B(α + 1) | δ ≥ Ω}
is closed under the operations that define B(α + 1). Clearly, Y is closed under +
and ϕ. If β ∈ Y and β < α + 1, then ψΩ (β) ≤ ψΩ (α) and hence ψΩ (β) ∈ Y .
Γ
(ii): α ∈ B(α + 1) yields ψΩ (α) < ψΩ (α + 1). Let ψΩ (α) < η < (ψΩ (α)) . Then
η∈
/ SC. By induction on η one therefore easily shows that η < ψΩ (α + 1). Together
Γ
with (i) this implies ψΩ (α + 1) = (ψΩ (α)) .
(iii) follows from Lemma 8.3(iii) since α ∈
/ B(α) yields B(α) ∩ [α, α + 1) = ∅. t
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Theorem: 8.9
(i) If ξ < ψΩ (Ω) then ξ < Γξ = ψΩ (ξ) < ψΩ (Ω).
(ii) ΓψΩ (Ω) = ψΩ (Ω).
(iii) If ψΩ (Ω) ≤ ξ ≤ Ω then ψΩ (ξ) = ψΩ (Ω).
Proof: Exercise.
Definition: 8.10 We write δ =N F ϕξ (η) if δ = ϕξ (η) and ξ, η < δ.
We write δ =N F ψΩ (α) if δ = ψΩ (α) and α ∈ B(α).
Note that by Lemma 8.3, δ =N F ψΩ (α) and δ =N F ψΩ (β) implies α = β.
Lemma: 8.11
(i) If β =N F β1 + . . . + βn and β ∈ B(α) then β1 , . . . , βn ∈ B(α).
(ii) If δ =N F ϕξ (η) ∈ B(α) then ξ, η ∈ B(α).
(iii) If δ =N F ψΩ (β) ∈ B(ρ) then β ∈ B(ρ) and β < ρ.
Proof: (i): Define
X := {β ∈ B(α) | if β =N F β1 + . . . + βn for some β1 , . . . , βn then β1 , . . . , βn ∈ B(α)}.
Show that X is closed under the operations that define B(α).
(ii): Define
Y := {β ∈ B(α) | if β =N F ϕξ (η) for some ξ, η then ξ, η ∈ B(α)}.
Show that Y is closed under the operations that define B(α).
(iii): ψΩ (β) ∈ B(ρ) implies ψΩ (β) < ψΩ (ρ) and hence β < ρ. As β ∈ B(β) we also
get β ∈ B(ρ).
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Remark: 8.12 It is essential to require
S that Ω ∈ B(α). If, instead of 0, Ω ∈ B(α),
one would require only 0 ∈ B(α), then α∈ON B(α) = σ, where σ is the least ordinal
such that Γσ = σ.
8.1
The ordinal representation system OT(Ω)
We will single out a set of ordinals that can be viewed as ordinal representation in
that all ordinals in it have a unique representation over the alphabet 0, Ω, +, ϕ, ψΩ ().
Definition: 8.13 The set OT(Ω) and Gα for α ∈ OT(Ω) are inductively defined
by the following clauses:
(R1) 0, Ω ∈ OT(Ω) and G0 = GΩ := 0.
(R2) If α =N F α1 + . . . + αn , n > 1 and α1 , . . . , αn ∈ OT(Ω) then α ∈ OT(Ω) and
Gα = max(Gα1 , . . . , Gαn ) + 1.
(R3) If α =N F ϕβ (δ), β, δ < Ω and β, δ ∈ OT(Ω) then α ∈ OT(Ω) and Gα =
max(Gβ, Gδ) + 1.
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(R4) If α =N F ω β , β > Ω and β ∈ OT(Ω) then α ∈ OT(Ω) and Gα = (Gβ) + 1.
(R5) If α =N F ψΩ (β), β ∈ OT(Ω) and β ∈ B(β) then α ∈ OT(Ω) and Gα = (Gβ)+1.
It follows from earlier results that any α ∈ OT(Ω) enters this set according to exactly
one of the rules (R1)-(R5) in exactly one way, and thus Gα is defined unambiguously.
Especially, OT(Ω) can be viewed as a set of terms which are composed of the symbols
0, Ω, +, ϕ, ψΩ in a unique way. What we are driving at next is a procedure that
enables us to decide for α, β ∈ OT(Ω) with α 6= β whether α < β or α > β solely
by inspection of their term representation. We also need a recipe to decide whether
an expression made up of the symbols 0, Ω, +, ϕ, ψΩ represents an ordinal of OT(Ω).
The main obstacle is raised by (R5) since we do not know how to deal with the
condition β ∈ B(β). This problems gives rise to the following definition.
Definition: 8.14 Inductive definition of Kα for α ∈ OT(Ω).
(K1) K0 = KΩ = ∅.
(K2) Kα = Kα1 ∪ . . . ∪ Kαn if α =N F α1 + . . . + αn where n > 1.
(K3) Kα = Kβ ∪ Kδ if α =N F ϕβ (δ).
(K4) Kα = Kβ ∪ {β} if α =N F ψΩ (β).
If X is a set of ordinals we write X < η to convey that ξ < η holds for all ξ ∈ X.
Note that Kα is always a finite set.
Lemma: 8.15 Let α ∈ OT(Ω). Then α ∈ B(ρ) if and only if Kα < ρ.
Proof: We proceed by induction on Gα.
If α =N F α1 + . . . + αn with n > 1 then:
α ∈ B(ρ) iff α1 , . . . , αn ∈ B(ρ) iff Kα1 ∪ . . . ∪ Kαn < ρ iff Kα < ρ,
using Lemma 8.11(i) and the induction hypothesis.
Likewise, if α =N F ϕη (β) then:
α ∈ B(ρ) iff η, β ∈ B(ρ) iff Kη ∪ Kβ < ρ iff Kα < ρ,
using Lemma 8.11(ii) and the induction hypothesis.
Now let α =N F ψΩ (β). Then:
α ∈ B(ρ) iff β ∈ B(ρ) ∧ β < ρ iff Kβ < ρ ∧ β < ρ iff Kα < ρ,
using Lemma 8.11(iii) and the induction hypothesis.
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Lemma: 8.16 If α ∈ OT(Ω) then ∀β ∈ Kα Gβ < Gα.
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Proof: Use induction on Gα.
Summarizing results from section 4 and this section we arrive at a primitive
recursive characterization of < on OT(Ω). Below we write α ∈ SC if α = Ω or
α =N F ψΩ (δ) for some δ.
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Lemma: 8.17 Let α, β ∈ OT(Ω). Then α < β holds if and only if one of the
following conditions is satisfied:
1. α = 0 and β 6= 0.
2. α =N F α0 + . . . + αn , β =N F β0 + . . . + βm , 0 < n < m and ∀i ≤ n αi = βi .
3. α =N F α0 + . . . + αn , β =N F β0 + . . . + βm , 0 < n, m and
∃i ≤ min(n, m)[∀j < i αj = βj ∧ αi < βi ].
4. α =N F α0 + . . . + αn , n > 0, β ∈ AP and α1 < β.
5. α ∈ AP, β =N F β0 + . . . + βn , n > 0 and α ≤ β1 .
6. α =N F ϕα1 (α2 ), β =N F ϕβ1 (β2 ), α1 < β1 and α2 < β.
7. α =N F ϕα1 (α2 ), β =N F ϕβ1 (β2 ), α1 = β1 and α2 < β2 .
8. α =N F ϕα1 (α2 ), β =N F ϕβ1 (β2 ), β1 < α1 and α < β2 .
9. α =N F ϕα1 (α2 ), α1 , α2 < β and β ∈ SC.
10. α ∈ SC, β =N F ϕβ1 (β2 ) and α ≤ max(β1 , β2 ).
11. α =N F ψΩ (α0 ), β =N F ψΩ (β0 ) and α0 < β0 .
12. α =N F ψΩ (α0 ) and β = Ω.
Proposition: 8.18 OT(Ω) ⊆ B(εΩ+1 ) ∩ εΩ+1 .
Proof: Use induction on Gα for α ∈ OT(Ω).
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9
KP goes infinite: LRS
A peculiarity of PA is that every object n of the intended model has a canonical
name in the language, namely, the nth numeral. It is not clear, though, how to
bestow a canonical name to each element of the set–theoretic universe. This is
where Gödel’s constructible universe L comes in handy. As L is “made” from the
ordinals it is pretty obvious how to “name” sets in L once one has names for ordinals.
These will be taken from OT(Ω). Henceforth, we shall restrict ourselves to ordinals
from OT(Ω).
Definition: 9.1 Up to know the basic symbols of our set-theoretic language have
been = and ∈. For technical reasons we would like to get rid of =. We simply define
a = b to be an abbreviation for
(∀x ∈ a) x ∈ b ∧ (∀x ∈ b) x ∈ a.
The axiom of extensionality then becomes a triviality. However, its role is taken
over by the equality axioms which we have not explicitly considered hitherto. The
role of extensionality is then played by the axiom
c = d ∧ c ∈ a → d ∈ a,
the unabbreviated version of which is
(∀x ∈ c)x ∈ d ∧ (∀x ∈ d)x ∈ c ∧ c ∈ a → d ∈ a.
Exercise: 9.2 Show that from the previous axiom one can deduce
c = d ∧ F (c) → F (d)
for any formula F (c).
Definition: 9.3 The set terms and their ordinal levels are defined inductively.
(i) For each α ∈ OT(Ω) ∩ Ω, there will be a set term Lα . Its ordinal level is
declared to be α.
(ii) If F (a, ~b ) is a set-theoretic formula, i.e. a formula of KP (whose free variables
are among the indicated) and ~s ≡ s1 , · · · , sn are set terms with levels < α,
then the formal expression
{x∈Lα | F (x, ~s )Lα }
is a set term of level α. Here F (x, ~s)Lα results from F (x, ~s) by restricting all
unbounded quantifiers to Lα .
A formula of RS is any expression of the form F (s1 , . . . , sn ), where F (a1 , . . . , an )
is a formula f KP with all free variables indicated and s1 , . . . , sn are set terms.
In the sequel, RS–formulae will be referred to just as formulae.
If A is a formula, then
k(A) := {α : Lα occurs in A }.
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Here any occurrence of Lα , i.e. also those inside of terms, has to be considered. For
a term s we set k(s) := k(s = s).
In what follows s, t, p, q, r, s1 , s2 , . . . will range over set terms. For a set term s
we shall notate the level of s by | s |. We also write s < t instead of | s | < | t |.
For terms s, t with | s | < | t | we set
◦
B(s)
if t ≡ {x ∈ Lβ | B(x)}
s∈t ≡
s∈
/ L0
if t ≡ Lβ .
The collection of set terms will serve as a formal universe for a theory LRS with
infinitary rules. The infinitary rule for the universal quantifier on the right takes the
form: From Γ ⇒ ∆, F (t) for all RS–terms t conclude Γ ⇒ ∆, ∀x F (x). There are
also rules for bounded universal quantifiers: From Γ ⇒ ∆, F (t) for all RS–terms
t with levels < α conclude Γ ⇒ ∆, (∀x ∈ Lα ) F (x). The corresponding rule for
introducing a universal quantifier bounded by a term of the form {x∈Lα : F (x, ~s)Lα }
is slightly more complicated. With the help of these infinitary rules it now possible to
give logical deductions of all axioms of KP with the exception of Bounded Collection.
The latter can be deduced from the rule of Σ-Reflection: From Γ ⇒ ∆, C conclude
Γ ⇒ ∆, ∃z C z for every Σ-formula C. The class of Σ-formulae is the smallest class
of formulae containing the bounded formulae which is closed under ∧, ∨, bounded
quantification and unbounded existential quantification. C z is obtained from C by
replacing all unbounded quantifiers ∃x in C by ∃x ∈ z.
The length and cut ranks of KP∞ -deductions will be measured by ordinals from
OT(Ω). If
KP ` F (u1 , . . . , ur )
then
LRS
Ω·m
Ω+n
B(s1 , . . . , sr )
holds for some m, n and all set terms s1 , . . . , sr ; m and n depend only on the
KP-derivation of B(~u).
Definition: 9.4 The inference rules of KP∞ include all the propositional inferences
of the sequent calculus (i.e., those pertaining to ∧, ∨, →, ¬) as well as the cut rule
(Cut). In addition, KP∞ has the following rules, where in (∈ R), (b∀ L) and (b∃ R)
it is also assumed that s < t):
Elementhood
◦
p ∈ t ∧ r = p, Γ ⇒ ∆ all p < t
r ∈ t, Γ ⇒ ∆
◦
(∈∞ )
Γ ⇒ ∆, s ∈ t ∧ r = s
Γ ⇒ ∆, r ∈ t
(∈ R)
Bounded Quantifiers
◦
s ∈ t → F (s), Γ ⇒ ∆
(∀x ∈ t) F (x), Γ ⇒ ∆
◦
Γ ⇒ ∆, p ∈ t → F (p) all p < t
(b∀ L)
Γ ⇒ ∆, (∀x ∈ t) F (x)
◦
p ∈ t ∧ F (p), Γ ⇒ ∆ all p < t
(∃x ∈ t) F (x), Γ ⇒ ∆
◦
(b∃∞ )
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Γ ⇒ ∆, s ∈ t ∧ F (s)
Γ ⇒ ∆, (∃x ∈ t) F (x)
(b∃ R)
(b∀∞ )
Unbounded Quantifiers
F (t), Γ ⇒ ∆
∀x F (x), Γ ⇒ ∆
Γ ⇒ ∆, F (p) for all p
(∀ L)
F (p), Γ ⇒ ∆ for all p
∃x F (x), Γ ⇒ ∆
Γ ⇒ ∆, ∀x F (x)
Γ ⇒ ∆, F (t)
(∃∞ )
Γ ⇒ ∆, ∃x F (x)
(∀∞ )
(∃ R)
Σ-Reflection
Γ ⇒ ∆, A
Γ ⇒ ∆, ∃x Ax
(Σ-Ref)
where A is a Σ-formula
Definition: 9.5 The rank of formulae and terms is determined as follows.
1. rk(Lα ) = ω · α.
2. rk({x∈Lα | F (x)}) = max{ω · α + 1, rk(F (L0 )) + 2}.
3. rk(s∈t) := max{rk(s) + 6, rk(t) + 1}.
4. rk(¬A) := rk(A) + 1.
5. rk(A ∧ B) = rk(A ∨ B) = rk(A → B) = max(rk(A), rk(B)) + 1.
6. rk((∃x∈t)F (x)) := rk((∀x∈t)F (x)) := max{rk(t), rk(F (L0 )) + 2}.
7. rk(∃xF (x)) := rk(∀xF (x)) := max{Ω, rk(F (L0 )) + 1}.
There is plenty of leeway in designing the actual rank of a formula.
Definition: 9.6 Let Pow(ON) = {X | X is a set of ordinals}.
A class function
H : Pow(ON) → Pow(ON)
will be called an operator if the following conditions are met for all X, X 0 ∈
Pow(ON):
(H0)
0 ∈ H(X).
(H1) For α =N F ω α1 + · · · + ω αn ,
α ∈ H(X) ⇐⇒ α1 , ..., αn ∈ H(X).
(In particular, (H1) implies that H(X) will be closed under + and σ 7→ ω σ ,
i.e., if α, β ∈ H(X), then α + β, ω α ∈ H(X).)
(H2)
X ⊆ H(X)
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(H3)
X 0 ⊆ H(X) =⇒ H(X 0 ) ⊆ H(X).
Note that an operator is monotone, i.e., if X 0 ⊆ X then X 0 ⊆ H(X) by (H2), and
hence H(X 0 ) ⊆ H(X) using (H3).
Definition: 9.7 (i) When f is a mapping f : ONk −→ ON, then H is said to
be closed under f , if, for all X∈Pow(ON) and α1 , . . . , αk ∈H(X),
f (α1 , . . . , αk )∈H(X).
(ii) α ∈ H := α ∈ H(∅);
s∈H := k(s) ⊆ H.
(iii) X ⊆ H := X ⊆ H(∅).
(iv) If Y is a set of ordinals we denote by H[Y ] the operator with
(H[Y ])(X) := H(Y ∪ X).
(v) For a set term s let H[s] denote the operator H[k(s)]
The next Lemma garners some simple properties of operators.
Lemma: 9.8 Let H be an operator, s be a set term and Y be a set of ordinals.
(i) H[Y ] and H[s] are operators.
(ii) Y ⊆ H =⇒ H[Y ] = H.
(iii) ∀X, X 0 ∈Pow(ON)[X 0 ⊆ X =⇒ H(X 0 ) ⊆ H(X)].
For a set of formulae Γ = {A1 , . . . , An } let k(Γ) = k(A1 ) ∪ . . . ∪ k(An ).
Definition: 9.9 We define the relation
H
α
ρ
Γ ⇒ ∆
by recursion on α by requiring that
k(Γ) ∪ k(∆) ∪ {α} ⊆ H(∅)
holds and one of the following conditions is satisfied:
1. Γ ⇒ ∆ is the result of a propositional inference (pertaining to one of the
αi
connectives ∧, ∨, →, ¬) with premisses Γi ⇒ ∆i and H ρ Γi ⇒ ∆i for some
αi < α.
α1
2. H ρ Γ, A ⇒ ∆ and H
with rk(A) < ρ.
α2
ρ
Γ ⇒ ∆, A for some α1 , α2 < α and formula A
3. Γ is of the form r ∈ t, Γ0 and
H[p]
αp
ρ
◦
p ∈ t ∧ r = p, Γ0 ⇒ ∆
holds for all p < t for some αp < α.
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4. ∆ is of the form ∆0 , r ∈ t and
◦
α0
ρ
H
Γ ⇒ ∆0 , s ∈ t ∧ r = s
holds for some s < t with | s | < α and some α0 < α.
5. Γ is of the form (∀x ∈ t) F (x), Γ0 and
H
α0
ρ
◦
s ∈ t → F (s), Γ0 ⇒ ∆
holds for some s < t with | s | < α and α0 < α.
6. ∆ is of the form ∆0 , (∀x ∈ t) F (x) and
H[p]
αp
ρ
◦
Γ ⇒ ∆, p ∈ t → F (p)
holds for all p < t for some αp < α.
7. Γ is of the form (∃x ∈ t) F (x), Γ0 and
H[p]
◦
αp
ρ
p ∈ t ∧ F (p), Γ0 ⇒ ∆
holds for all p < t for some αp < α.
8. ∆ is of the form ∆0 , (∃x ∈ t) F (x) and
H
α0
ρ
◦
Γ ⇒ ∆0 , s ∈ t ∧ F (s)
holds for some s < t with | s | < α and some α0 < α.
9. Γ is of the form ∀x F (x), Γ0 and
H
α0
ρ
F (s), Γ0 ⇒ ∆
holds for some s with | s | < α and α0 + 2 < α.
10. ∆ is of the form ∆0 , ∀x F (x) and
H[p]
αp
ρ
Γ ⇒ ∆, F (p)
holds for all p for some αp + 2 < α.
11. Γ is of the form ∃x F (x), Γ0 and
αp
ρ
H[p]
F (p), Γ0 ⇒ ∆
holds for all p for some αp + 2 < α.
12. ∆ is of the form ∆0 , ∃x F (x) and
H
α0
ρ
Γ ⇒ ∆0 , F (s)
holds for some s with | s | < α and some α0 + 2 < α.
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13. α ≥ Ω and ∆ is of the form ∆0 , ∃z Az , where A is a Σ-formula, and
H
α0
ρ
Γ ⇒ ∆0 , A
holds for some α0 + 1 < α.
Lemma: 9.10
and
(i) If Γ0 ⊆ Γ, ∆0 ⊆ ∆, k(Γ), k(∆) ⊆ H, α ∈ H, α0 ≤ α, ρ0 ≤ ρ
α0
ρ0
H
then
α
ρ
H
(ii) If H
α
ρ
Γ0 ⇒ ∆0
Γ ⇒ ∆.
Γ ⇒ ∆, (∀x ∈ Lβ )F (x) , γ ∈ H and γ ≤ β then H
α
ρ
Γ ⇒ ∆, (∀x ∈ Lγ )F (x)
Proof: (i) is proved by a straightforward induction on α0 .
For (ii) we use induction on α. the only interesting case is when (∀x ∈ Lγ )F (x)
was the principal formula of the last inference which would have been (b∀)∞ . So we
have
αp
H[s] ρ Γ ⇒ ∆, (∀x ∈ Lβ )F (x), p ∈
/ L0 ∧ F (p)
for all p < β, where αp < α. By the induction hypothesis we get
H[s]
αp
ρ
Γ ⇒ ∆, (∀x ∈ Lγ )F (x), p ∈
/ L0 ∧ F (p)
for all p < γ and thus, via another (b∀)∞ inference, we get the desired result.
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Lemma: 9.11 If k(s) ⊆ H, α ∈ H and α > 0 then
H
Proof: We have H[p]
αp
0
α
0
⇒ s∈
/ L0 .
◦
p ∈ L0 ∧ p = s ⇒ for all p < 0 for some αp < 0 (since
there ain’t any such p). Hence, via an inference (∈)∞ we get H[p]
α
from which we get H 0 ⇒ s ∈
/ L0 via (¬R).
0
0
s ∈ L0 ⇒ ,
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Lemma: 9.12 The inversions (i)-(viii) of RA∗ of Lemma 5.10 concerning propositional logic also hold for RS. In addition the following inversions hold for RS.
(i) If H
α
ρ
r ∈ t, Γ ⇒ ∆ then H[p]
α
ρ
◦
p ∈ t ∧ r = p, Γ ⇒ ∆ holds for all p < t.
◦
α
α
ρ
Γ ⇒ ∆, p ∈ t → F (p) holds for all
α
α
ρ
Γ ⇒ ∆, p ∈ t ∧ F (p) holds for all
(ii) If H ρ Γ ⇒ ∆, (∀x ∈ t)F (x) then H[p]
p < t.
(iii) If H ρ (∃x ∈ t)F (x), Γ ⇒ ∆ then H[p]
p < t.
◦
(iv) If H
α
ρ
Γ ⇒ ∆, ∀x F (x) then H[s]
α
ρ
Γ ⇒ ∆, F (s) holds for all s.
(v) If H
α
ρ
∃x F (x), Γ ⇒ ∆ then H[s]
α
ρ
F (s), Γ ⇒ ∆ holds for all s.
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Proof: All are straightforward by induction on α.
α
Lemma: 9.13 (Reduction) Let ρ = |C| =
6 Ω. If H ρ Γ, C ⇒ ∆ and H
then
α#α#β#β
H ρ
Γ, Ξ ⇒ ∆, Θ .
β
ρ
Ξ ⇒ Θ, C ,
Proof: The proof is by induction on α#α#β#β and very similar to Lemma 5.11.
We only look at two cases where C and was the principal formula of the last inference
in both derivations. It is essential to notice that C is not the principal formula of
an inference (Σ-Ref) since |C| =
6 Ω.
Case 1: The first is when C is of the form r ∈ t. Then we have
αp
ρ
H[p]
◦
Γ, C, p ∈ t ∧ r = p ⇒ ∆
for all p < t with αp < α and
H
β0
ρ
◦
Ξ ⇒ Θ, C, s ∈ t ∧ r = s
for some β0 < β and term s < t with | s | < β.
Since k(s) ⊆ H we also have that H = H[s].
By the induction hypothesis we obtain
◦
H
αs #αs #β#β
ρ
Γ, Ξ, s ∈ t ∧ r = s ⇒ ∆, Θ
H
α#α#β0 #β0
ρ
Γ, Ξ ⇒ ∆, Θ, s ∈ t ∧ r = s .
and
◦
◦
α#α#β#β
Γ, Ξ ⇒ ∆, Θ .
Cutting out s ∈ t ∧ r = s gives H ρ
Case 2: The second case is when C is of the form (∀x ∈ t)A(x) Then we have
H
α1
ρ
◦
Γ, C, s ∈ t → A(s) ⇒ ∆
for some α1 < α and s < t with | s | < α. And we also have
H[s]
βs
ρ
◦
Γ ⇒ ∆, C, s ∈ t → A(s)
for some βs < β and s < t with | s | < β. Since k(s) ∈ H we have H[s] = H. By the
induction hypothesis we thus get
α1 #α1 #β#β
ρ
Γ, Ξ, s ∈ t → A(s) ⇒ ∆, Θ
H
α#α#βs #βs
ρ
Γ, Ξ ⇒ ∆, Θ, s ∈ t → A(s) .
and
◦
◦
H
Cutting out s ∈ t → A(s) gives H
◦
α#α#β#β
ρ
Γ, Ξ ⇒ ∆, Θ .
Theorem: 9.14 (First Cut Elimination Theorem)
α
4α
If H δ+1 Γ ⇒ ∆ and δ 6= Ω then H δ Γ ⇒ ∆ .
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Proof: Use induction on α and the previous Lemma.
Theorem: 9.15 (Predicative cut elimination) Let H be closed under ϕ.
α
If H ρ+ων Γ ⇒ ∆ , Ω ∈
/ [ρ, ρ + ω ν [ and ν ∈ H, then
ϕν (α)
ρ
H
Γ ⇒ ∆.
Proof: By main induction on ν and subsidiary induction on α. The assertion holds
for ν = 0 by the First Cut Elimination Theorem 9.14 since ρ 6= Ω. Now suppose
ν > 0. There will be a last inference (I) with premisses Γi ⇒ ∆i . Suppose the
αi
inference was not a cut or a cut of rank < ρ. We then have H[i] ρ+ων Γi ⇒ ∆i for
some αi < α. By the subsidiary induction hypothesis we have H[i]
ϕν (αi )
ρ
Γi ⇒ ∆i .
ϕν (α)
ρ
Applying the same inference (I) yields H
Γ ⇒ ∆.
Now suppose the last inference was a cut with cut formula C such that ρ ≤
|C| < ρ + ω ν . Then there exist ν0 < ν and n < ω such that |C| < ρ + ω ν0 · n. After
performing a cut with C we have
ϕν (α)
H
ρ+ω ν0 ·n
Γ ⇒ ∆.
We also have ϕν0 (ϕν (α)) = ϕν (α). Therefore by n-fold application of the main
ϕν (α)
ρ
induction hypothesis we obtain H
Γ ⇒ ∆.
t
u
Lemma: 9.16 (Bounding Lemma) Let B be a Σ-formula and A be a Π-formula.
Suppose α ≤ β < Ω and β ∈ H.
(i) If H
(ii) If H
α
ρ
α
ρ
Γ ⇒ ∆, B then
H
α
ρ
Γ ⇒ ∆, B Lβ .
H
α
ρ
Γ, ALβ ⇒ ∆ .
Γ, A ⇒ ∆ then
Proof: (i) Use induction on α. Note that the deductions cannot contain any
inference (Σ-Ref) since α < Ω.
Note that if B is not the principal formula of the last inference then the assertion
follows readily from the induction hypothesis. So let’s assume that B was the
principal formula of the last inference. If B is a ∆0 formula or of either form
B0 ∨ B1 , B0 ∧ B1 , (∀x ∈ t)F (x), or (∃x ∈ t)F (x) then the assertion follows readily
from the induction hypothesis. So suppose B is of the form ∃xF (x). Then we have
H
α0
ρ
Γ ⇒ ∆, B, F (s)
for some α0 + 2 < α and a term s with | s | < α. Inductively we have
(∗) H
We also have
(∗∗) H
α0
ρ
Γ ⇒ ∆, B Lβ , F (s)Lβ .
α0 +1
ρ
Γ ⇒ ∆, B Lβ , s ∈
/ L0
78
by Lemma 9.11. Thus from (∗) and (∗∗) we get
H
α0 +2
ρ
/ L0 ∧ F (s)Lβ
Γ ⇒ ∆, B Lβ , s ∈
α0 +2
via (∧R). The latter is the same as H ρ
| s | < β, and hence, using (b∃R), we get H
79
◦
Γ ⇒ ∆, B Lβ , s ∈ Lβ ∧ F (s)Lβ since
α
Lβ
.
t
u
ρ Γ ⇒ ∆, B
10
Impredicative Cut Elimination
The usual cut elimination procedure works unless the cut formulae have been introduced by Σ-reflection rules. The obstacle to pushing cut elimination further is
exemplified by the following scenario:
δ
Ω
Γ ⇒ ∆, C
ξ
Γ ⇒ ∆, ∃z C z
Ω
···
RefΣ
ξs
Ω
Ξ, C s ⇒ Λ · · · (| s |< Ω)
ξ
Ξ, ∃z C z ⇒ Λ
Ω
α
Γ, Ξ ⇒ ∆, Λ
Ω+1
(∃L)
(Cut)
In order to be able to remove these critical cuts, i.e. cuts which were introduced by (Σ-Ref), we have to forgo arbitrary operators. We shall need operators H
such that an H–controlled derivation that satisfies certain extra conditions can be
“collapsed” into a derivation with much smaller ordinal labels.
From now on we will identify ON with B(ΩΓ ). All operators are therefore
supposed to just act on subsets of B(ΩΓ ).
Definition: 10.1 The operator Hη for η < εΩ+1 is defined by
\
Hη (X) = {B(β) | X ⊆ B(β) ∧ η < β}.
Lemma: 10.2
(i) Hη is an operator.
(ii) η < η 0 =⇒ Hη (X) ⊆ Hη0 (X).
(iii) Hη is closed under ϕ and ψΩ η + 1.
Proof: (i): X ⊆ Hη (X) follows by definition. If X 0 ⊆ Hη (X), then, for any β > η
such that X ⊆ B(β), we have X 0 ⊆ B(β), and therefore Hη (X 0 ) ⊆ B(β), hence
Hη (X 0 ) ⊆ Hη (X).
So far we have verified (H0), (H2) and (H3). As to (H1), suppose α =N F ω α1 +
. . . + ω αn . We have to show
α ∈ Hη (X) iff α1 , . . . , αn ∈ Hη (X).
But this is a consequence of
α ∈ B(β) iff α1 , . . . , αn ∈ B(β)
which holds by Lemma 8.11(i).
(ii) is obvious. (iii) follows from the fact that the sets B(β) with β > η are closed
under ϕ and ψΩ η + 1.
t
u
Lemma: 10.3 Suppose η ∈ Hη . Define β̂ := η + ω Ω+β .
(i) If α ∈ Hη then α̂, ψΩ (α̂) ∈ Hα̂ .
(ii) If α0 ∈ Hη and α0 < α then ψΩ (αˆ0 ) < ψΩ (α̂).
80
Proof: Obviously, Hη (∅) = B(η +1). From α, η ∈ B(η +1) we obtain α̂ ∈ B(α̂), and
hence ψΩ (α̂) ∈ B(α̂+1) = Hα̂ (∅). This shows (i). Now suppose α0 ∈ Hη and α0 < α.
By the preceding argument we then have ψΩ (αˆ0 ) ∈ B(α̂), thus ψΩ (αˆ0 ) < ψΩ (α̂). t
u
Lemma: 10.4 (Persistence) Let δ ∈ H.
(i) If H
α
ρ
Γ ⇒ ∆, ∀x F (x) then H
α
ρ
Γ ⇒ ∆, (∀x ∈ Lδ )F (x) .
(ii) If H
α
ρ
∃x F (x), Γ ⇒ ∆ then H
α
ρ
(∃x ∈ Lδ )F (x), Γ ⇒ ∆ .
Proof: (i): We proceed by induction on α. The only interesting case is when the
last inference was (∀)∞ . Thus
H[s]
αs
ρ
Γ ⇒ ∆, ∀x F (x), F (s)
holds for all s for some αs + 2 < α. Inductively we have
H[s]
and hence
H[s]
αs
ρ
αs +1
ρ
◦
Γ, s ∈ Lδ ⇒ ∆, (∀x ∈ Lβ )F (x), F (s)
◦
Γ ⇒ ∆, (∀x ∈ Lβ )F (x), s ∈ Lδ → F (s)
for all | s | < β. Thus, via (b∀)∞ we conclude that H
(ii) is similar.
α
ρ
Γ ⇒ ∆, (∀x ∈ Lδ )F (x) .
t
u
Theorem: 10.5 (Collapsing and Impredicative Cut Elimination) Let Γ be set
of Π-formulae and ∆ be a set of Σ-formulae. Suppose that η ∈ Hη . Then
Hη
α
Γ ⇒ ∆
Ω+1
implies Hα̂
ψΩ (α̂)
ψΩ (α̂)
Γ ⇒ ∆
where α̂ = η + ω Ω+α .
This result can also be established for the intuitionistic version of RS provided
one adds the extra assumption that all formulae in Γ have rank at most Ω.
Proof: We proceed by induction on α.
Case 0: If the last inference was propositional then the assertion follows easily from
the induction hypothesis.
Case 1: Suppose the last inference was (b∀)∞ . Then a formula (∀x ∈ t)F (x)
appears in ∆ and
H[p]
◦
αp
Γ ⇒ ∆, p ∈ t → F (p)
Ω+1
holds for all p < t for some αp < α. Since k(t) ⊆ H we have k(t) ⊆ B(η + 1) and
thus | t | < ψΩ (η + 1). As a result, | p | < ψΩ (η + 1) and therefore k(p) ⊆ H holds
◦
for all p < t, and hence H[p] = H for all p < t. The formula p ∈ t → F (p) might
not be a Σ-formula but F (p) is a Σ-formula since (∀x ∈ t)F (x) is. Using inversion
(Lemma 9.12) we have
H
αp
Ω+1
◦
Γ, p ∈ t ⇒ ∆, F (p)
81
(61)
for all p < t Thus we can apply the induction hypothesis to (61), yielding
ψΩ (αˆp )
Hαˆp
ψΩ (αˆp )
◦
Γ, p ∈ t ⇒ ∆, F (p)
and hence
ψΩ (αˆp )
Hαˆp +1
ψΩ (αˆp )
◦
Γ ⇒ ∆, p ∈ t → F (p)
(62)
for all p < t. As ψΩ (αˆp ) + 1 < ψΩ (α̂) holds by Lemma 10.3(ii), we can apply an
inference (b∀)∞ to get Hα̂
ψΩ (α̂)
ψΩ (α̂)
Γ ⇒ ∆.
Case 3: Suppose the last inference was (Σ-Ref). Then ∆ contains a formula ∃z Az ,
where A is a Σ-formula and
H
α0
Ω+1
Γ ⇒ ∆, A
for some α0 < α. By the induction hypothesis we have
Hαˆ0
ψΩ (αˆ0 )
ψΩ (αˆ0 )
Γ ⇒ ∆, A .
Using the Bounding Lemma 9.16 we get
Hαˆ0
ψΩ (αˆ0 )
ψΩ (αˆ0 )
Lψ
Γ ⇒ ∆, A
Ω
(αˆ0 )
.
Via an inference (∃R) we get
Hαˆ0
ψΩ (αˆ0 )+2
ψΩ (αˆ0 )
Γ ⇒ ∆, ∃z Az .
Since ψΩ (αˆ0 ) + 2 < ψΩ (α̂), by Lemma 10.3, and ∃z Az is in ∆, we also have
Hα̂
ψΩ (α̂)
ψΩ (α̂)
Γ ⇒ ∆.
Case 4: Suppose the last inference was a cut. Then there exists a formula C with
rk(C) ≤ Ω and α0 < α such that
H
H
α0
Ω+1
α0
Ω+1
Γ, C ⇒ ∆ ;
(63)
Γ ⇒ ∆, C .
(64)
Case 4.1: rk(C) < Ω. Then we can apply the induction hypothesis to both (63)
and (64) so that
Hαˆ0
Hαˆ0
ψΩ (αˆ0 )
ψΩ (αˆ0 )
ψΩ (αˆ0 )
ψΩ (αˆ0 )
Γ, C ⇒ ∆ ;
(65)
Γ ⇒ ∆, C .
(66)
Since k(C) ⊆ Hη this implies rk(C) < ψΩ (η + 1). Thus applying a cut to (65) and
(66) yields Hα̂
ψΩ (α̂)
ψΩ (α̂)
Γ ⇒ ∆.
82
Case 4.2: rk(C) = Ω. Then C is of the form Qx F (x) with Q ∈ {∃, ∀} and F (L0 )
being ∆0 . Let’s first suppose that C is ∃x F (x). Then we can apply the induction
hypothesis to (64) and we get
Hαˆ0
ψΩ (αˆ0 )
ψΩ (αˆ0 )
Γ ⇒ ∆, C .
(67)
Using the Persistence Lemma 10.4 and the fact that ψΩ (αˆ0 ) ∈ Hαˆ0 (invoking Lemma
10.3(i)) we infer from (63) that
Hαˆ0
α0
Ω+1
Γ, (∃x ∈ LψΩ (αˆ0 ) )F (x) ⇒ ∆ .
(68)
Since (∃x ∈ LψΩ (αˆ0 ) )F (x) is ∆0 the induction hypothesis can be applied to (68),
yielding
Hα1
ψΩ (α1 )
ψΩ (α1 )
Γ, (∃x ∈ LψΩ (αˆ0 ) )F (x) ⇒ ∆ ,
(69)
where α1 = αˆ0 + ω Ω+α0 . Since α1 < η + ω Ω+α = α̂ and rk((∃x ∈ LψΩ (αˆ0 ) )F (x)) <
ψΩ (α̂) hold, cutting with (67) and (69) furnishes Hα̂
ψΩ (α̂)
ψΩ (α̂)
Γ ⇒ ∆.
If C is ∀x F (x) the argument is similar.
11
t
u
Interpreting KP in RS
Theorem: 11.1 (Interpretation Theorem) If KP ` A where A is sentence then
there exist m < ω such that
Ω·ω m
H0 Ω+m A .
Proof: The proof is too long to be incorporated here.
Corollary: 11.2
t
u
(i) If A is a Σ sentence of KP and KP ` A then
LψΩ (εΩ+1 ) |= A.
(ii) If KP ` C where C is a sentence of the form ∀x∃y F (x, y) with F (a, b) being
a Σ formula, then
LψΩ (εΩ+1 ) |= C.
(iii) There is no ordinal < ψΩ (εΩ+1 ) that satisfies (i).
(iv) k KP k= ψΩ (εΩ+1 ).
Proof: (i): Suppose KP ` A. By Theorem 11.1 we find m < ω such that
H0
Ω·ω m
Ω+m
A.
We can assume that m > 1. Using the First Cut Elimination Theorem 9.14 m-1times we get
H0
σ0
Ω+1
83
A
(70)
where σ0 := ωm−1 (Ω · ω m ). Note that to (70) we can apply Impredicative Cut
Elimination 10.5, and hence, since 0 + ω Ω+σ0 = ω σ0 ,
Hσ1
ψΩ (σ1 )
ψΩ (σ1 )
A
(71)
where σ1 = ω σ0 . By the Bounding Lemma 9.16 it follows that
Hσ1
σ2
σ2
ALσ2
(72)
where σ2 = ψΩ (σ1 ). By Predicative Cut Elimination 9.15 we conclude from (71)
that
Hσ1
ϕσ2 (σ2 )
0
ALσ2 .
(73)
As the derivation from (73) contains no inference (Σ-Ref) one then shows by induction on ϕσ2 (σ2 ) that all sequents appearing in the derivation are true in Lσ2 on the
standard interpretation.
Obviously, ϕσ2 (σ2 ) < ψΩ (εΩ+1 ). As A is a Σ-formula it follows that LψΩ (εΩ+1 ) |=
B.
(ii) follows from (i) and (iii) using Theorem 2.1 from M. Rathjen: Fragments of
Kripke-Platek set theory with infinity, in: P. Aczel, H. Simmons, S. Wainer (eds.):
Proof Theory (Cambridge University Press, Cambridge, 1992) 251-273.
(iii) requires a well-ordering proof in KP.
(iv) follows from the fact that PA + TI(ψΩ (εΩ+1 )) proves the consistency of KP and
a cunning argument involving Löb’s Theorem.
t
u
ψΩ (εΩ+1 ) is also known as the Bachmann-Howard ordinal.
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