Download The Epsilon Calculus

The Epsilon Calculus
Georg Moser
Institut für Mathematische Logik
Universität Münster
http://www.math.uni-muenster.de/logik/org/staff/moserg/
Richard Zach
Department of Philosophy
University of Calgary
http://www.ucalgary.ca/∼rzach/
August 26 & 27, 2003
CSL’03, Vienna
ε-Calculus
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What is the Epsilon Calculus?
The ε-calculus is a formalization of logic without quantifiers
but with the ε-operator.
If A(x) is a formula, then εx A(x) is an ε-term.
Intuitively, εx A(x) is an indefinite description: εx A(x) is
some x for which A(x) is true.
ε can replace ∃: ∃x A(x) ⇔ A(εx A(x))
Axioms of ε-calculus:
• Propositional tautologies
• (Equality schemata)
• A(t) → A(εx A(x))
Predicate logic can be embedded in ε-calculus.
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Why Should You Care?
• Epsilon calculus is of significant historical interest.
– Origins of proof theory
– Hilbert’s Program
• Alternative basis for fruitful proof-theoretic research.
– Epsilon Theorems and Herbrand’s Theorem: proof
theory without sequents
– Epsilon Substitution Method: yields functionals, e.g.,
` ∀x∃y A(x, y)
∀n : ` A(n, f (n))
• Interesting Logical Formalism
– Trade logical structure for term structure.
– Suitable for proof formalization.
– Choice in logic.
– Inherently classical (but see work of Bell, DeVidi, Fitting,
Mostowski).
– Expressive power?
• Other Applications:
– Use of choice functions in provers (e.g., HOL, Isabelle).
– Applications in linguistics (choice functions, anaphora).
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Outline
1. Historical Remarks
2. Overview of Results
3. The Epsilon-Calculus: Syntax and Semantics
4. The Epsilon Theorems
Tomorrow:
5. The First Epsilon Theorem
6. The Second Epsilon Theorem and Herbrand’s Theorem
7. Generalizations of the Epsilon Theorems
8. (Intermediate) Conclusion
9. Hilbert’s “Ansatz” for the Epsilon Substitution Method
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Historical Remarks
The epsilon calculus was first introduced by Hilbert in
1922, as the basis for his formulation of mathematics for which
Hilbert’s Program was supposed to be carried out.
Motivation: Logical choice function; ε-terms represent
“ideal elements” in proofs.
Original work in proof theory (pre-Gentzen) concentrated
on ε-calculus and ε-substitution method (Ackermann, von
Neumann, Bernays)
First correct proof of Herbrand’s Theorem via Epsilon
Calculus
Epsilon substitution method used by Kreisel for nocounterexample interpretation leading to work on analysis of
proofs by Kreisel, Luckhardt, Kohlenbach.
Recent work on ordinal analysis of subsystems of analysis
by Arai, Avigad, Buchholz, Mints, Tupailo, Tait.
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The Epsilon Calculus: Syntax
Language of the Elementary Calculus LEC:
• Free variables: a, b, c, . . .
• Bound variables: x, y, z, . . .
• Constant and function symbols: f , g, h, . . . with arities
ar( f ), . . .
• Predicate symbols: P, Q, R, . . . with arities ar(P), . . .
• Equality: =
• Propositional connectives: ∧, ∨, →, ¬
Language of the Predicate Calculus LPC:
• Quantifiers: ∀, ∃
Language of the Epsilon Calculus Lε:
• Epsilon: ε
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The Epsilon Calculus: Syntax (cont’d)
Semi-formulas and Semi-terms:
1. Any free variable a is a semi-term.
2. Any bound variable x is a semi-term.
3. If f is a function symbol with ar( f ) = 0, then f is a semiterm.
4. If f is a function symbol with ar( f ) = n > 0, and t1, . . . , tn
are semi-terms, then f (t1, . . . ,tn) is a semi-term.
5. If P is a predicate symbol with ar(P) = n > 0, and t1, . . . , tn
are terms, then P(t1, . . . ,tn) is an (atomic) semi-formula.
6. If A and B are semi-formulas, then ¬A, A ∧ B, A ∨ B and
A → B are semi-formulas.
7. If A(a) is a semi-formula containing the free variable a and
x is a bound variable not occurring in A(a), then ∀x A(x) and
∃x A(x) are semi-formulas.
8. If A is a semi-formula containing the free variable a and x
is a bound variable not occurring in A(a), then εx A(x) is a
semi-term (an ε-expression).
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The Epsilon Calculus: Syntax (cont’d)
Terms and formulas:
Terms and formulas are defined exactly as above except
that clause (2) is dropped. In other words: a semi-term is
a term and a semi-formula a formula if it contains no bound
variables without a matching ∀, ∃, or ε.
Subterms and sub-semi-terms:
1. The only sub(semi)terms of a free variable a is a itself. It
has no immediate sub(semi)terms.
2. A bound variable x has no subterms or immediate subsemi-terms. Its only subsemiterm is x itself.
3. If f (t1, . . .tn) is a semi-term, then its immediate sub-semiterms are t1, . . . , tn, and its immediate subterms are those
among t1, . . . , tn which are terms, plus the immediate
subterms of those among t1, . . . , tn which aren’t terms.
Its sub-semi-terms are f (t1, . . . ,tn) and the sub-semi-terms
of t1, . . . , tn; its subterms are those of its sub-semi-terms
which are terms.
4. If P(t1, . . . ,tn) is an atomic semi-formula, then its immediate
sub-semi-terms are t1, . . . , tn. Its immediate subterms are
those among t1, . . . , tn which are terms, plus the immediate
subterms of those among t1, . . . , tn which aren’t terms. Its
sub(semi)terms are the sub(semi)terms of t1, . . . , tn.
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5. If A is a semi-formula of the form B ∧ C, B ∨ C, B → C, ¬B,
∀x B or ∃x B, then its immediate sub(semi) formulas and its
sub(semi)formulas are those of B and C.
6. If εx A(x) is an ε-expression then its immediate sub(semi)
formulas and its sub(semi)formulas are those of A(x).
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The Epsilon Calculus: Syntax (cont’d)
Epsilon matrices
An ε-term εx A(x) is an ε-matrix—or simply, a matrix—if all
terms occurring in A are free variables, each of which occurs
exactly once.
Denote ε-matrices as εx A(x; a1, . . . , ak ) where the variables
a1, . . . , ak displayed are all the free variables occurring in it.
Two matrices εx A(x; a1, . . . , ak ), εx A(x; b1, . . . , bk ) that differ
only in the indicated tuples of variables a and b, respectively,
are considered to be equal. The set of all matrices is denoted
Mat.
Corresponding to each ε-term εx A(x) there exists a unique
matrix εx A(x; a1, . . . , ak ) and a unique sequence t1, . . . ,tk of
terms such that εx A(x,t1, . . . ,tk ) = εx A(x).
Example: εx A(s, εy B(y), εzC(x,t))
| {z }
e
Its matrix is: εx A(a, b, εzC(x, c))
The matrix of εx A(x) is obtained by replacing all immediate
subterms of εx A(x) by distinct new free variables. In this newly
obtained term we replace distinct occurrences of the same
variable by different variables.
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The Epsilon Calculus: Intensional Semantics
An intensional choice function Φ for a set X is a
function Φ : 2X × Mat × X <ω → X so that for S ⊆ X, a matrix
εx A(x; a1, . . . , an) and d1, . . . , dn ∈ S, Φ(S, εx A(x; a1, . . . , an), hd1, . . . , dni) ∈
/
S if S 6= 0.
An intensional ε-structure M = h|M|, (·)M, ΦMi consist of
/ an interpretation function (·)M, and an
a domain |M| 6= 0,
intensional choice function ΦM for |M|, where
1. If ar( f ) = 0, then f M ∈ |M|
2. If ar( f ) = n > 0, then f M : |M|n → |M|
3. If ar(P) = n > 0, then PM ⊆ |M|n
A variable assignment s for M is a function s : FV → |M|.
We write s ∼c s0 if s(a) = s0(a) for all free variables a other than
c.
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The Epsilon Calculus: Intensional Semantics
(cont’d)
The value t M,s of a term t and the satisfaction relation
M, s |= A is defined by:
1.
2.
3.
4.
5.
6.
7.
8.
If a is a free variable, then aM,s = s(a).
If f is a constant symbol, then f M,s = f M.
If f (t1, . . . ,tn) is a term, then ( f (t1, . . . ,tn))M,s = f M(t1M,s, . . . ,tnM,s).
If P(t1, . . . ,tn) is an atomic formula, then M, s |= P(t1, . . . ,tn)
iff ht1M,s, . . . ,tnM,si ∈ PM.
If A and B are formulas, then:
• M, s |= A ∧ B iff M, s |= A and M, s |= B.
• M, s |= A ∨ B iff M, s |= A or M, s |= B.
• M, s |= A → B iff M, s 6|= A or M, s |= B.
• M, s |= ¬A iff M, s 6|= A.
If ∃x A(x) is a formula and c a free variable not in A(x), then
M, s |= ∃x A(x) iff M, s0 |= A(c) for some s0 ∼c s.
If ∀x A(x) is a formula and c a free variable not in A(x), then
M, s |= ∀x A(x) iff M, s0 |= A(c) for all s0 ∼c s.
If e is an ε-term with matrix εx A(x; a1, . . . , an) so that e =
εx A(x;t1, . . . ,tn), and
X = {d ∈ |M| : M, s0 |= A(c;t1, . . . ,tn), s0 ∼c s, s0(c) = d},
then
eM,s = ΦM(X, εx A(x; a1, . . . , an), ht1M,s, . . . ,tnM,si).
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Axiomatization of the Epsilon Calculus: Axioms
Let AxEC be the set of formulas containing all formulas
which are either substitution instances of propositional
tautologies, or substitution instances of
a
=
a
a=b
→
(A(a) → A(b)).
AxPC is AxEC together with all substitution instances of
A(a)
→
∃x A(x)
∀x A(x)
→
A(a).
Any substitution instance of
A(a) → A(εx A(x))
is called a critical formula.
We obtain AxεEC and AxεPC by adding the critical formulas
to AxEC and AxPC, respectively.
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Axiomatization of the Epsilon Calculus:
Deductions
A deduction in EC or ECε is a sequence A1, . . . , An of
formulas such that each Ai is either in AxEC or AxεEC or follows
from A j , Ak with j, k < i by modus ponens:
B, B → C ` C
A deduction in PC or PCε is a sequence A1, . . . , An of
formulas such that each Ai is either in AxPC or AxεPC or follows
from A j , Ak with j, k < i by modus ponens: B, B → C ` C, or
follows from A j with j < i by generalization:
B → A(a)
`
B → ∀x A(x)
A(a) → B
`
∃x A(x) → B
A formulas A is called deducible (in EC, ECε, PC, PCε),
` A, if there is a deduction (in AxEC, AxεEC, AxPC, AxεPC,
respectively) which has A as its last formula.
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Extensional Epsilon Calculus
Alternative semantics: Choice function maps just sets to
elements:
ΦM(S) ∈ S
Then if e = εx A(x,t1, . . . ,tn)
eM,s = ΦM({d ∈ |M| : M, s0 |= A(c;t1, . . . ,tn), s0 ∼c s0, s0(c) = d})
In extensional ε-semantics, equivalent ε-terms have same
value, i.e., it makes valid the ε-extensionality axiom
∀x(A(x) ↔ B(x)) → εx A(x) = εx B(x)
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Embedding PC in ECε
The epsilon operator allows the treatment of quantifiers in
a quantifier-free system: using ε terms it is possible to define
∃x and ∀x as follows:
∃x A(x)
⇔
A(εx A(x))
∀x A(x)
⇔
A(εx ¬A(x))
Define Aε by:
xε = x, aε = a
[εx A(x)]ε = εx A(x)ε
f (t1, . . . ,tn)ε = f (t1ε, . . . ,tnε), P(t1, . . . ,tn)ε = P(t1ε, . . . ,tnε).
(A ∧ B)ε = Aε ∧ Bε, (A ∨ B)ε = Aε ∨ Bε, (A → B)ε = Aε → Bε,
(¬A)ε = ¬Aε.
5. (∃x A(x))ε = Aε(εx Aε(x)0).
6. (∀x A(x))ε = Aε(εx ¬Aε(x)0).
1.
2.
3.
4.
where Aε(x)0 is Aε(x) with all variables bound by quantifiers or
ε’s in Aε(x) renamed by new bound variables (to avoid collision
of bound variables when εx Aε(x) is substituted into Aε(x) where
x may be in the scope of a quantifier or epsilon binding a
variable occurring in Aε(x).
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Embedding PC in ECε: Examples
∃x(P(x)
∨
∀y Q(y))ε =
=
[P(x) ∨ ∀y Q(y)]ε
{x ← εx [P(x) ∨ ∀y Q(y)]ε}
[P(x) ∨ ∀y Q(y)]ε = P(x) ∨ Q(εy ¬Q(y))
| {z }
e1
=
P(x) ∨ Q(εy ¬Q(y)) {x ← εx [P(x) ∨ Q(εy ¬Q(y))]}
| {z }
| {z }
e1
e1
|
{z
}
e2
=
P(εx [P(x) ∨ Q(εy ¬Q(y))]) ∨ Q(εy ¬Q(y))
| {z }
| {z }
e1
e1
|
{z
}
e2
[∃x
∃y
A(x, y)]ε =
=
[∃y A(x, y)]ε
{x ← εx [∃y A(x, y)]ε}
[∃y A(x, y)]ε = A(x, εy A(x, y))
| {z }
e1 (x)
=
A(x, εy A(x, y)){x ← εx A(x, εz A(x, z))}
| {z }
|
{z
}
e1 (x)
=
e2
A(εx A(x, εz A(x, z)), εy A(εx A(x, εz A(x, z)), y))
|
{z
}
|
{z
}
e2
e
|
{z2
}
e1 (e2 )
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Embedding PC in ECε (cont’d)
Prop. If PCε ` A then ECε ` Aε.
Translations of quantifier axioms provable from critical
formulas:
[A(t) → ∃x A(x)]ε = Aε(t) → Aε(εx A(x)ε)
Suppose PCε ` B → A(a), and a does not occur in B. By IH, we
have a proof π in ECε of Bε → A(a)ε. Replacing a everywhere
in π by εx ¬A(x) results in a proof of
[B → ∀x A(x)]ε = Bε → Aε(εx ¬A(x)ε).
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The Epsilon Theorems
First Epsilon Theorem. If A is a formula without bound
variables (no quantifiers, no epsilons) and PCε ` A then EC `
A.
Extended First Epsilon Theorem. If ∃x1 . . . ∃xnA(x1, . . . , xn)
is a purely existential formula containing only the bound
variables x1, . . . , xn, and
PCε ` ∃x1 . . . ∃xnA(x1, . . . , xn),
then there are terms ti j such that
EC `
_
A(ti1, . . . ,tin).
i
Second Epsilon Theorem.
PCε ` A then PC ` A.
If A is an ε-free formula and
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Degree and Rank
Degree of an ε-Term
• deg(εx A(x)) = 1 if A(x) contains no ε-subterms.
• If e1, . . . , en are all immediate ε-subterms of A(x), then
deg(εx A(x)) = max{deg(e1), . . . , deg(en)} + 1.
Rank of an ε-Expression
An ε-expression e is subordinate to εx A if e is a proper
sub-expression of A and contains x.
• rk(e) = 1 if no sub-ε-expression of e is subordinate to e.
• If e1, . . . , en are all the ε-expressions subordinate to e, then
rk(e) = max{rk(e1), . . . , rk(en)} + 1
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Examples
P(εx [P(x) ∨ Q(εy ¬Q(y))]) ∨ Q(εy ¬Q(y))
| {z }
| {z }
e1
e1
|
{z
}
e2
deg(e1) = 1, deg(e2) = 2
rk(e1) = rk(e2) = 1
A(εx A(x, εz A(x, z)), εy A(εx A(x, εz A(x, z)), y))
|
{z
}
|
{z
}
e2
e
|
{z2
}
e1 (e2 )
deg(e2) = 1, deg(e1(e2)) = 2
rk(e2) = 2, rk(e1(e2)) = 1
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Rank of Critical Formulas and Derivations
Rank of a critical formula A(t) → A(εx A(x)) is rk(εx A(x)).
Rank of a derivation rk(π): maximum rank of its critical
formulas.
Critical ε-term of a derivation: ε-term e so that A(t) → A(e)
is a critical formula.
Order of a derivation o(π, r) wrt. rank r: number of different
critical ε-terms of rank r.
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The First Epsilon Theorem
(Proof for case without =)
Suppose PCε `π E and E contains no bound variables. We
show that EC ` E by induction on the rank and degree of π.
First, w.l.o.g. we assume π is actually a derivation in ECε .
Since E contains no bound variables, E ε = E.
Second, w.l.o.g. we assume π doesn’t contain any free
variables (replace free variables by new constants—may be
resubstituted later).
Lemma. Let e be a critical ε-term of π of maximal degree
among the critical ε-terms of maximal rank. Then there is πe
with end formula A so that rk(πe) ≤ rk(π), deg(πe) ≤ deg(π) and
o(πe, rk(e)) = o(π, rk(e)) − 1.
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The First Epsilon Theorem: Main Lemma
Proof. Construct πe as follows:
1. Suppose A(t1) → A(e), . . . , A(tn) → A(e) are all the critical
formulas belonging to e. For each critical formula
A(ti) → A(e),
we obtain a derivation
πi ` A(ti) → E :
• Replace e everywhere it occurs by ti. Every critical
formula A(t) → A(e) belonging to e turns into a formula
of the form B → A(ti).
• Add A(ti) to the axioms. Now every such formula is
derivable using the propositional tautology
A(ti) → (B → A(ti)) ,
and modus ponens.
• Apply the deduction theorem for the propositional
calculus to obtain πi.
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The First Epsilon Theorem: Main Lemma
2. Obtain a derivation π0 of ¬A(ti) → E by:
V
• Add ¬A(ti) to the axioms. Now every critical formula
A(ti) → A(e) belonging to e is derivable using the
propositional tautology ¬A(ti) → (A(ti) → A(e)).
• Apply the deduction theorem.
V
3. Combine the proofs
πi ` A(ti) → E ,
and
π `
0
^
¬A(ti) → E ,
to get πe ` E (case distinction)
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Why is this correct?
Verify that the resulting derivation is indeed a derivation in
ECε with the required properties.
We started with critical formulas of the form
A(ti) → A(e) .
Facts:
• The proof π0 does not contain any critical formulas
belonging to e. Hence e is no longer a critical ε-term
in π0. All other critical formulas (and the critical ε-terms
they belong to) remain unchanged. Thus o(π0, rk(e)) =
o(π, rk(e)) − 1.
• In the construction of πi, we substituted e by t throughout
the proof. Such uniform substitutions of a term by another
are proof-preserving.
• Replacing e by ti in A(e) indeed results in A(ti), since e
cannot occur in A(x)—else e = εx A(x) would be a proper
subterm of itself, which is impossible.
• If e appears in another critical formula B(s) → B(εy B(y)), we
have three cases.
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Case I
Case: e occurs only in s.
Replacing e by ti results in a critical formula B(s0) →
B(εy B(y)).
The new critical critical formula belongs to the same ε-term
as the original formula.
Hence o(πi, rk(e)) = o(π, rk(e)) − 1.
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Case II
Case: e may occur in B(y) and perhaps also in s, but
contains neither s nor εy B(y).
In other words, the critical formula has the form
B0(s0(e), e) → B0(εy B0(y, e), e).
But then the ε-term belonging to this critical formula
e0 = εy B0(y, e) ,
is of higher degree than e.
By our assumptions, this implies that rk(εy B0(y, e)) < rk(e).
Replacing e by ti results in a different critical formula
B0(s0(ti),ti) → B0(εy B0(y,ti),ti) ,
belonging to the ε-term εy B0(y,ti) which has the same rank as
e0 and hence a lower rank than e itself.
Hence again o(πi, rk(e)) = o(π, rk(e)) − 1.
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Case III
Case: e does contain s or εy B(y).
Then e is of the form e0(s) or e0(εy B(y)), and
B(a) is really of the form B0(e0(a)) where e0(a) is an ε-term
of the same rank as e.
Then εy B(y) has the form εy B0(e0(y)), to which the εexpression e0(y) is subordinated.
But then εy B0(e0(y)) has higher rank than e0(y), which has
the same rank as e. This cannot happen.
Finally the proof of the lemma follows: In all of the cases
considered one ε-critical term of rk(e) was removed and other
ε-critical terms of rk(e) remained equal. Thus o(πe, rk(e)) =
o(π, rk(e)) − 1 holds.
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The First Epsilon Theorem: Proof
By induction on rk(π).
If rk(π) = 0, there is nothing to prove (no critical formulas).
If rk(π) > 0 and the order of π wrt. rk(π) is m, then mfold application of the lemma results in a derivation π0 of rank
< rk(π).
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The Extended First Epsilon Theorem
Theorem. If ∃x1 . . . ∃xk A(x1, . . . , xk ) is a purely existential
formula containing only the bound variables x1, . . . , xk , and
PCε ` ∃x1 . . . ∃xk A(x1, . . . , xk ),
then there are terms ti j such that
EC `
_
A(ti1, . . . ,tik ).
i
Consider proofs in PCε of ∃x1 . . . ∃xk A(x1, . . . , xk ), where
A(a1, . . . , ak ) contains no bound variables.
Employing embedding we obtain a derivation π of
A(s1, . . . , sk ), where s1, . . . , sk are terms (containing ε’s).
Proof Sketch.
We employ the same sequence of
elimination steps as in the proof of the First Epsilon Theorem.
The difference being that now the end-formula A(s1, . . . , sk ) may
contain ε-terms.
Hence the first elimination step transform the end-formula
into a disjunction.
A(s01, . . . , s0k ) ∨ . . . ∨ A(sn1, . . . , snk ) .
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The Second Epsilon Theorem
Theorem. If A is an ε-free formula and PCε ` A then PC `
A.
Assume A has the form
∃x∀y∃zB(x, y, z) ,
with B(x, y, z) quantifier-free and no other than the indicated
variables occur in A.
Herbrand Normal Form. Suppose A = ∃x∀y∃zB(x, y, z). If
f is a new function symbol, then the Herbrand normal form AH
of A is ∃x∃zB(x, f (x), z).
Lemma. Suppose PCε ` A. Then PCε ` AH .
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Second Epsilon Theorem: Proof
The Strong First Epsilon Theorem yields:
There are ε-free terms ri, si so that
_
EC `
B(ri, f (ri), si)
(1)
i
We now can replace the ti by new free variables ai and obtain
from (1), that
_
B(ri, ai, si) ,
(2)
i
is deducible in EC.
Then the original prenex formula A can be obtained
from (2) if we employ the following rules (deducible in PC)
(µ) : F ∨ G(t) ` F ∨ ∃y G(y)
(ν) : F ∨ G(a) ` F ∨ ∀z G(z), provided a appears only in G(a)
at the displayed occurrences.
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Corollaries
Conservative Extension.
Due to the Second Epsilon
Theorem the Epsilon Calculus (with equality) is a conservative
extension of pure predicate logic.
Equivalence.
Due to the Embedding Lemma we have
PCε ` A implies ECε ` Aε. Due to the Second Epsilon Theorem
we obtain ECε ` Aε implies PCε ` A.
Herbrand’s Theorem.
Assume A = ∃x∀y∃zB(x, y, z).
Iff PCε ` A, then there are terms ri, si such that EC `
W
i B(ri , f (ri ), si ).
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Generalizations
First Epsilon Theorem.
Let A be a formula without
bound variables (no quantifiers, no epsilons) but possible
including =. Then
PCε ∪ Ax ` A implies EC ∪ Ax ` A ,
where Ax includes instances of quantifier-free (and ε-free)
axioms.
Extended First Epsilon Theorem.
Let ∃xA(x) be a
purely existential formula (possibly containing =). Then
PCε ∪ Ax ` ∃xA(x) implies
EC ∪ Ax `
_
A(ti1, . . . ,tin) ,
i
where Ax is defined as above.
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Generalizations (cont’d)
Second Epsilon Theorem.
(possibly containing =) and
If A is an ε-free formula
PCε ∪ Ax ` A implies PC ∪ Ax ` A ,
where Ax includes instances of ε-free axioms.
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(Intermediate) Conclusion
Some facts in favour of the Epsilon Calculus:
• The input parameter for the proof of Herbrand’s Theorem is
the collection of critical formulas C used in the derivation.
E.g. this gives a bound depending only on C .
• The Epsilon Calculus allows a condensed representation
of proofs.
Why: Assume ECε ` Aε. Then there exists a tautology of
the form
^
(Bi(t j ) → Bi(εx Bi(x))) → Aε .
(3)
i, j
Thus as soon as the critical formulas Bi(t j ) → Bi(εx Bi(x))
are known, we only need to verify that (3) is a tautology to
infer that Aε is provable in ECε.
• Formalization of proofs should be simpler in the Epsilon
Calculus.
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A bluffer’s guide to Hilbert’s “Ansatz”
Assume we work within number theory and let N denote
the standard model of number theory.
(For conciseness we ignore induction.)
• The initial substitution S0: Assign the constant function 0
to all ε-terms (by assigning the constant function 0 to all
ε-matrices).
• Assume the substitution Sn has already been defined.
Define Sn+1: Pick a false critical axiom, e.g.
P(t) → P(εx P(x)) .
(False means wrt. to N and the current substitution Sn.)
• Let z ∈ N denote the value of t under Sn. Then the next
substitution Sn+1 is obtained by assigning the value z to
εx P(x).
Note that the critical axiom P(t) → P(εx P(x)) is true wrt. to N
and the current substitution Sn+1.
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Peano Arithmetic: Results
1-consistency.
derivable in PAε is true.
Every purely existential formula
Provable Recursive Functions. The numerical content
of proofs of purely existential formulas in PAε is extractable.
Put differently: The provable recursive functions of PAε are
exactly the < ε0-recursive functions.
Assume PAε ` ∀x∃yA(x, y) with A(a, b) quantifier-free and
without free variables other than the shown. Then we can find
a < ε0-recursive function f such that ∀xA(x, f (x)) holds.
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Peano Arithmetic: Results (cont’d)
Non-counter example interpretation.
Let
∃x∀y∃zA(a, x, y, z)
be deducible in PAε such that only the indicated free variable
a occurs. Let ∃x∃zA(a, x, f (x), z) denote the Herbrand normal
form of A.
Then there exists < ε0-recursive functionals G and H such
that for all functions f ,
A(n, G( f , n), f (G), H( f , n)) ,
holds.
The transformation ()ε : Lε → LPC defined yesterday, can
be employed to show that Peano Arithmetic embeds into PAε:
Then if PA ` A, then PAε ` Aε.
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