Download Model theory of second order logic

Introduction
Second order characterizable structures
Model theory of second order logic
Lecture 1
Jouko Väänänen1 ,2
1 Department
of Mathematics and Statistics, University of Helsinki
2 ILLC,
University of Amsterdam
August 2011
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Outline
1
Introduction
Introduction
Number theory
Analysis
Set theory
2
Second order characterizable structures
Second order characterizable structures
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
Outline
1
Introduction
Introduction
Number theory
Analysis
Set theory
2
Second order characterizable structures
Second order characterizable structures
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
Our framework
Usual set theoretic framework.
In the third lecture a more general approach.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
First order logic: for every element a of the domain of the
model ...
Second order logic: for every subset of (or relation on) the
domain of the model ...
Third order logic: for every family of subsets of (or relation
between relations on) the domain of the model ...
....
Sort logic: for every superset M 0 of the domain M and
every relation on M 0 ...
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
First order logic: for every element a of the domain of the
model ...
Second order logic: for every subset of (or relation on) the
domain of the model ...
Third order logic: for every family of subsets of (or relation
between relations on) the domain of the model ...
....
Sort logic: for every superset M 0 of the domain M and
every relation on M 0 ...
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
First order logic: for every element a of the domain of the
model ...
Second order logic: for every subset of (or relation on) the
domain of the model ...
Third order logic: for every family of subsets of (or relation
between relations on) the domain of the model ...
....
Sort logic: for every superset M 0 of the domain M and
every relation on M 0 ...
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
First order logic: for every element a of the domain of the
model ...
Second order logic: for every subset of (or relation on) the
domain of the model ...
Third order logic: for every family of subsets of (or relation
between relations on) the domain of the model ...
....
Sort logic: for every superset M 0 of the domain M and
every relation on M 0 ...
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
First order logic: for every element a of the domain of the
model ...
Second order logic: for every subset of (or relation on) the
domain of the model ...
Third order logic: for every family of subsets of (or relation
between relations on) the domain of the model ...
....
Sort logic: for every superset M 0 of the domain M and
every relation on M 0 ...
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
M is characterizable by a sentence (or theory) φ in some
logic if for all M0 :
M0 ∼
= M ⇐⇒ M0 |= φ.
M characterizable in first order logic iff M is finite.
Infinite M: Cannot characterize in first order logic ⇒ model
theory.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
M is characterizable by a sentence (or theory) φ in some
logic if for all M0 :
M0 ∼
= M ⇐⇒ M0 |= φ.
M characterizable in first order logic iff M is finite.
Infinite M: Cannot characterize in first order logic ⇒ model
theory.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
M is characterizable by a sentence (or theory) φ in some
logic if for all M0 :
M0 ∼
= M ⇐⇒ M0 |= φ.
M characterizable in first order logic iff M is finite.
Infinite M: Cannot characterize in first order logic ⇒ model
theory.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
M is characterizable by a sentence (or theory) φ in some
logic if for all M0 :
M0 ∼
= M ⇐⇒ M0 |= φ.
M characterizable in first order logic iff M is finite.
Infinite M: Cannot characterize in first order logic ⇒ model
theory.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
Can an infinite M be characterized in second order logic?
For many: Yes.
For all e.g. countable: Depends on set theory.
Depends on model theoretic properties of M.
Model theory of second order logic is very different from
model theory of first order logic.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
Can an infinite M be characterized in second order logic?
For many: Yes.
For all e.g. countable: Depends on set theory.
Depends on model theoretic properties of M.
Model theory of second order logic is very different from
model theory of first order logic.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
Can an infinite M be characterized in second order logic?
For many: Yes.
For all e.g. countable: Depends on set theory.
Depends on model theoretic properties of M.
Model theory of second order logic is very different from
model theory of first order logic.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
Can an infinite M be characterized in second order logic?
For many: Yes.
For all e.g. countable: Depends on set theory.
Depends on model theoretic properties of M.
Model theory of second order logic is very different from
model theory of first order logic.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
M = (M, R1 , ..., Rn , f1 , ..., fm , c1 , ..., ck ).
Can an infinite M be characterized in second order logic?
For many: Yes.
For all e.g. countable: Depends on set theory.
Depends on model theoretic properties of M.
Model theory of second order logic is very different from
model theory of first order logic.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
There is a more general notion of a structure: Henkin
structure (M, G), where G is a set of subsets of and
relations on M, for the range of second order quantifiers,
such that certain comprehension axioms are satisfied.
Every usual structure M gives rise to a Henkin structure
(M, P(M) ∪ P(M 2 ) ∪ ...), called a full structure.
Some results about all full models hold also for Henkin
models.
Sometimes we cannot get a full model and have to settle
with a Henkin model.
“Henkin model" is an auxiliary concept.
Typically, full models “lurk" inside clouds of Henkin models.
Truth in Henkin models can be axiomatized, truth in full
models not.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
There is a more general notion of a structure: Henkin
structure (M, G), where G is a set of subsets of and
relations on M, for the range of second order quantifiers,
such that certain comprehension axioms are satisfied.
Every usual structure M gives rise to a Henkin structure
(M, P(M) ∪ P(M 2 ) ∪ ...), called a full structure.
Some results about all full models hold also for Henkin
models.
Sometimes we cannot get a full model and have to settle
with a Henkin model.
“Henkin model" is an auxiliary concept.
Typically, full models “lurk" inside clouds of Henkin models.
Truth in Henkin models can be axiomatized, truth in full
models not.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
There is a more general notion of a structure: Henkin
structure (M, G), where G is a set of subsets of and
relations on M, for the range of second order quantifiers,
such that certain comprehension axioms are satisfied.
Every usual structure M gives rise to a Henkin structure
(M, P(M) ∪ P(M 2 ) ∪ ...), called a full structure.
Some results about all full models hold also for Henkin
models.
Sometimes we cannot get a full model and have to settle
with a Henkin model.
“Henkin model" is an auxiliary concept.
Typically, full models “lurk" inside clouds of Henkin models.
Truth in Henkin models can be axiomatized, truth in full
models not.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
There is a more general notion of a structure: Henkin
structure (M, G), where G is a set of subsets of and
relations on M, for the range of second order quantifiers,
such that certain comprehension axioms are satisfied.
Every usual structure M gives rise to a Henkin structure
(M, P(M) ∪ P(M 2 ) ∪ ...), called a full structure.
Some results about all full models hold also for Henkin
models.
Sometimes we cannot get a full model and have to settle
with a Henkin model.
“Henkin model" is an auxiliary concept.
Typically, full models “lurk" inside clouds of Henkin models.
Truth in Henkin models can be axiomatized, truth in full
models not.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
A rough correspondence
first order logic
first order logic with
the well-ordering quantifier
first order logic with
the Härtig quantifier
second order logic
sort logic
Boolean valued sort
(or second order) logic
Jouko Väänänen
∆KPU−Inf
-part of set theory
1
∆1 -part of set theory
∆1 (Cd)-part of set theory
∆2 -part of set theory
set theory
Ω-logic
(Ongoing work with D. Ikegami)
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
Outline
1
Introduction
Introduction
Number theory
Analysis
Set theory
2
Second order characterizable structures
Second order characterizable structures
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
Second order axioms of number theory, P 2
Dedekind analyzed and Peano axiomatized number theory in
1888 and 1889. Also Peirce 1881.
Constant symbol c for zero, function symbol S for
successor function.
∀y ¬Sy = c,
∀y ∀z(Sy = Sz → y = z),
∀X ((Xc ∧ ∀y (Xy → XSy )) → ∀yXy ) (Induction)
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
Second order axioms of number theory, P 2
Theorem (Peirce 1881 (?), Dedekind 1888, Peano 1889 (?))
There is, up to isomorphism, only one model of P 2 , namely the
standard model (N, S, 0).
Proof.
If (M, s, a) |= P 2 , interpret Y as {a, sa, ssa, sssa, ...}.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
Note
In first order logic there is no theory T such that (N, S, 0)
is, up to isomorphism, the only model of T .
Addition, multiplication, etc can be defined in second order
logic from 0 and S, by recursion. Not so in first order logic.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
Outline
1
Introduction
Introduction
Number theory
Analysis
Set theory
2
Second order characterizable structures
Second order characterizable structures
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
An axiomatization of real numbers, R 2
Karl Weierstrass brought the theory of the real numbers
(from geometry) to algebra and set theory.
David Hilbert axiomatized the ordered field of real numbers
in 1902.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
Real numbers, R 2
Second order axioms of real numbers, R 2 :
Constant symbols for zero and one, function symbols +, ×,
relation symbol <.
•
•
•
Field axioms for +, ·, 0 and 1
x <y →x +z <y +z
0 <x, 0 < y → 0 < xy
•
∀X
∃yXy ∧ ∃x∀y (Xy → y ≤ x)) →
∃x∀y ∀z(Xz → z ≤ y ) ↔ x ≤ y
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
Characterization of the reals
Theorem
There is, up to isomorphism, only one model of R 2 , namely
(R, +, ·, 0, 1, ≤).
Proof.
Given two models, find integers, then rationals in both. Map
rationals to rationals. Extend to isomorphism using
completeness.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
First order versus second order
In second order logic natural numbers can be defined on
R. Not so in first order logic.
Tarski 1948: The ordered field of reals is decidable in first
oder logic. Not so in second order logic.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
Outline
1
Introduction
Introduction
Number theory
Analysis
Set theory
2
Second order characterizable structures
Second order characterizable structures
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
Second order axioms of set theory, ZFC 2 , Zermelo
1908
Usual ZFC axioms except replacement and separation are
written in second order form. A finite set of axioms ZFC 2
obtains.
Models of ZFC 2 are well-founded and the order-type of
their ordinals is a strongly inaccessible cardinal.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
Second order axioms of set theory, ZFC 2
Theorem (Zermelo 1930)
For any strongly inaccessible cardinal > ω there is, up to
isomorphism, exactly one model of ZFC 2 with the set of
ordinals of order-type κ, namely (Vκ , ∈).
Proof.
Transfinite induction on the rank. Suppose a well-founded
(M, E) is a model of ZFC 2 of inaccessible height κ. W.l.o.g.
(M, E) is a transitive structure (M, ∈), hence M ⊆ Vκ . Easily, M
is supertransitive (Suppose X ⊆ x ∈ M. Then X ⊆ M, as M is transitive. By
separation, X ∈ M). Hence M = Vκ .
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
In second order set theory mathematical questions outside
set theory are decided.
Continuum Hypothesis CH ∀A ⊆ R(|A| ≤ |N| ∨ |A| = |R|) is
decided by ZFC 2 , but we do not (yet) know in which way ...
... just as we do not (yet) know how (V , ∈) decides CH in
set theory.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
In second order set theory mathematical questions outside
set theory are decided.
Continuum Hypothesis CH ∀A ⊆ R(|A| ≤ |N| ∨ |A| = |R|) is
decided by ZFC 2 , but we do not (yet) know in which way ...
... just as we do not (yet) know how (V , ∈) decides CH in
set theory.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Introduction
Number theory
Analysis
Set theory
In second order set theory mathematical questions outside
set theory are decided.
Continuum Hypothesis CH ∀A ⊆ R(|A| ≤ |N| ∨ |A| = |R|) is
decided by ZFC 2 , but we do not (yet) know in which way ...
... just as we do not (yet) know how (V , ∈) decides CH in
set theory.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
Outline
1
Introduction
Introduction
Number theory
Analysis
Set theory
2
Second order characterizable structures
Second order characterizable structures
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
Second order characterizable structures
A model M is second order characterizable if there is a
second order sentence θM such that for all M 0 :
M 0 |= θM ⇐⇒ M 0 ∼
= M.
Characterizability by a theory is defined similarly.
We already know many second order characterizable
structures. However, there are only countably many such.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
Second order characterizable structures
A model M is second order characterizable if there is a
second order sentence θM such that for all M 0 :
M 0 |= θM ⇐⇒ M 0 ∼
= M.
Characterizability by a theory is defined similarly.
We already know many second order characterizable
structures. However, there are only countably many such.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
Second order characterizable structures
A model M is second order characterizable if there is a
second order sentence θM such that for all M 0 :
M 0 |= θM ⇐⇒ M 0 ∼
= M.
Characterizability by a theory is defined similarly.
We already know many second order characterizable
structures. However, there are only countably many such.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
1
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Second order characterizable structures
Natural numbers (N, +, ·, 0, 1, <)
The field of real numbers (R, +, ·, 0, 1, <)
The field of complex numbers (C, +, ·, 0, 1)
The Euclidean space (Rn , || · ||)
The Banach space (`p , || · ||p )
The Hilbert space (Lp , || · ||p )
The Euclidean geometry, consisting of points and lines
The ordered set (ωn , <)
The pure set (ℵn )
The Boolean algebra (P(ω), ∩, ∪, ∅, −)
Any finite structure
Any recursive countable structure
The random graph
The free group of ℵn generators
The level of the cumulative hierarchy (Vω+n , ∈)
The level of the cumulative hierarchy (Vωn , ∈)
The level of the cumulative hierarchy (Vκ , ∈), where κ is the first inaccessible
(Mahlo, weakly compact, Ramsey) cardinal.
Every consistent recursive first order theory has a second order characterizable
model in each second order characterizable infinite cardinality.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
Non-second order characterizable structures
Not second order characterizable:
(R, +, ×, 0, 1, <, ≺), where ≺ is a well-order, assuming
large cardinals. (Martin-Steel 1989)
(N, <, P), where P is the set of Gödel numbers of valid
second order sentences.
As to (N, <, P), we give a proof of a more general result:
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
Non-second order characterizable structures
Not second order characterizable:
(R, +, ×, 0, 1, <, ≺), where ≺ is a well-order, assuming
large cardinals. (Martin-Steel 1989)
(N, <, P), where P is the set of Gödel numbers of valid
second order sentences.
As to (N, <, P), we give a proof of a more general result:
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
We give two proofs of:
Theorem
There is no second order characterizable structure M such that
the set of Gödel numbers of valid second order sentences is
Turing-reducible to truth in M.
Proof.
The theory of any second order characterizable structure is ∆2
(see below). The set of Gödel numbers of valid second order
sentences is Π2 -complete (see below). A Π2 -complete set
cannot be Turing reducible to a ∆2 -set, by the Hierarchy
Theorem of the Levy-hierarchy.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
Figure: The hierarchy of second order characterizable structures
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
We prove two fact that we used in the above proof:
Theorem
If A is a second order characterizable structure, then the theory
of A is ∆2 -definable.
Theorem
The set of second order φ such that |= φ, is Π2 -complete.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
Proof elements
L a finite vocabulary, A a second order characterizable
L-structure.
σ = the conjunction of a large finite part of ZFC.
Sut(M) a Π1 -formula which says that M is supertransitive.
Voc(x) = the definition of “x is a vocabulary".
SO(L, x) = the set-theoretical definition of the class of
second order L-formulas.
Str(L, x) = the set-theoretical definition of L-structures.
Sat(A, φ) = the inductive truth-definition of second order
logic written in the language of set theory.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
Two ways to say y is true in the z-structure x
P1 (z, x, y ) = Voc(z) ∧ Str(z, x) ∧ SO(z, y ) ∧
∃M(z, x, y ∈ M ∧ σ (M) ∧ Sut(M)
∧(Sat(z, x, y ))(M) ) (Σ2 )
P2 (z, x, y ) = Voc(z) ∧ SO(z, y ) ∧ Str(z, x) ∧
∀M((z, x, y ∈ M ∧ σ (M) ∧ Sut(M))
→ (Sat(z, x, y ))(M) ) (Π2 ).
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
ZFC ` ∀z∀x∀y (P1 (z, x, y ) ↔ P2 (z, x, y ))
A |= φ iff ∃x(P1 (L, x, φ) ∧ P1 (L, x, θA )) iff
∀x(P1 (L, x, θA ) → P2 (L, x, φ)). This shows that the second
order theory of a second order characterizable A is ∆2 .
If L is a vocabulary and φ a second order L-sentence, then
|= φ ⇐⇒ ∀x(Str(L, x) → P1 (L, x, φ)). This shows that “φ
is valid" is Π2 . Now we show it is Π2 -complete.
Jouko Väänänen
Model theory of second order logic
Introduction
Second order characterizable structures
Second order characterizable structures
Suppose ∃x∀yP(x, y , n) is a Σ2 -predicate. Let φn be a
second order sentence the models of which are, up to
isomorphism, exactly the models (Vα , ∈), where α = iα
and (Vα , ∈) |= ∃x∀yP(x, y , n). If ∃x∀yP(x, y , n) holds, we
can find a model for φn by means of the Levy Reflection
principle. On the other hand, suppose φn has a model.
W.l.o.g. it is of the form (Vα , ∈). Let a ∈ Vα such that
(Vα , ∈) |= ∀yP(a, y , n). Since in this case Hα = Vα ,
(Hα , ∈) |= ∀yP(a, y , n), where Hα is the set of sets of
hereditary cardinality < α. By another application of the
Levy Reflection Principle we get (V , ∈) |= ∀yP(a, y , n), and
we have proved ∃x∀yP(x, y , n).
Jouko Väänänen
Model theory of second order logic