Model Theory of Second Order Logic

Transcription

1 Lecture 2 1, 2 1 Department of Mathematics and Statistics University of Helsinki 2 ILLC University of Amsterdam March 2011

2 Outline Second order characterizable structures 1 Second order characterizable structures 2 Significance of the

3 Outline Second order characterizable structures 1 Second order characterizable structures 2 Significance of the

4 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. This is the case even if we study just the valid Σ 1 1-sentences of second order logic. Proof. The theory of any second order characterizable structure is 2 (see below). The set of Gödel numbers of valid second order sentences is Π 2 -complete (see below). A Π 2 -complete set cannot be Turing reducible to a 2 -set, by the Hierarchy Theorem of the Levy-hierarchy.

5 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. This is the case even if we study just the valid Σ 1 1-sentences of second order logic. Proof. The theory of any second order characterizable structure is 2 (see below). The set of Gödel numbers of valid second order sentences is Π 2 -complete (see below). A Π 2 -complete set cannot be Turing reducible to a 2 -set, by the Hierarchy Theorem of the Levy-hierarchy.

6 We prove two facts 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 (even just Σ 1 1 ) φ such that = φ, is Π 2 -complete.

7 Proof elements Second order characterizable structures L a finite vocabulary, A a second order characterizable L-structure. σ = the conjunction of a large finite part of ZFC. Sut(M) a Π 1 -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.

8 Two ways to say y is true in the z-structure x P 1 (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) ) (Σ 2 ) P 2 (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) ) (Π 2 ).

9 ZFC z x y(p 1 (z, x, y) P 2 (z, x, y)) A = φ iff x(p 1 (L, x, φ) P 1 (L, x, θ A )) iff x(p 1 (L, x, θ A ) P 2 (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) P 1 (L, x, φ)). This shows that φ is valid" is Π 2. Now we show it is Π 2 -complete.

10 Suppose x yp(x, y, n) is a Σ 2 -predicate. Let φ n be a Π 1 1-second order sentence the models of which are, up to isomorphism, exactly the models (V α, ), where α = ℶ α and (V α, ) = x yp(x, y, n). If x yp(x, y, n) holds, we can find a model for φ n by means of the Levy Reflection principle. On the other hand, suppose φ n has a model. W.l.o.g. it is of the form (V α, ). Let a V α such that (V α, ) = yp(a, y, n). Since in this case H α = V α, (H α, ) = yp(a, y, n), where H α 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).

11 Suppose x yp(x, y, n) is a Σ 2 -predicate. Let φ n be a Π 1 1-second order sentence the models of which are, up to isomorphism, exactly the models (V α, ), where α = ℶ α and (V α, ) = x yp(x, y, n). If x yp(x, y, n) holds, we can find a model for φ n by means of the Levy Reflection principle. On the other hand, suppose φ n has a model. W.l.o.g. it is of the form (V α, ). Let a V α such that (V α, ) = yp(a, y, n). Since in this case H α = V α, (H α, ) = yp(a, y, n), where H α 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).

12 An alternative proof Suppose = φ is Turing-reducible to truth in A, and A is second order characterizable. Then truth in any second order characterizable structure is Turing-reducible to truth in A. There is a second order characterizable structure B of size 2 A. Hence truth in B is reducible to truth in A. We show that this is impossible: Theorem If A and B are any infinite second order characterizable structures such that 2 A B, then truth in B is not Turing reducible to truth in A.

13 Let A = κ, B = λ. Both are second order characterizable, and so are B = (λ P(κ), λ, <, P(κ), κ, π, N) and A = (κ, <, π, N), where π is a bijection of κ κ onto κ. We show B is not Turing-reducible to A. L = the vocabulary of B, L L that of A. Suppose for all second order L-sentences φ: B = φ A = φ. Can write a second order L-sentence Θ(x, y) such that for all L -formulas φ(x) and any n N B = Θ( φ, n) A = φ(n), thus A = φ(n) A = Θ( φ, n). Let ψ(x) be the L -formula which says Θ(a, a) for the natural number" a, in the formal sense, that is the value of x. Let k = ψ(x). Now κ = Θ(k, k) A = ψ(k) A = (Θ(k, k)), a contradiction.

14 Let A = κ, B = λ. Both are second order characterizable, and so are B = (λ P(κ), λ, <, P(κ), κ, π, N) and A = (κ, <, π, N), where π is a bijection of κ κ onto κ. We show B is not Turing-reducible to A. L = the vocabulary of B, L L that of A. Suppose for all second order L-sentences φ: B = φ A = φ. Can write a second order L-sentence Θ(x, y) such that for all L -formulas φ(x) and any n N B = Θ( φ, n) A = φ(n), thus A = φ(n) A = Θ( φ, n). Let ψ(x) be the L -formula which says Θ(a, a) for the natural number" a, in the formal sense, that is the value of x. Let k = ψ(x). Now κ = Θ(k, k) A = ψ(k) A = (Θ(k, k)), a contradiction.

15 Let A = κ, B = λ. Both are second order characterizable, and so are B = (λ P(κ), λ, <, P(κ), κ, π, N) and A = (κ, <, π, N), where π is a bijection of κ κ onto κ. We show B is not Turing-reducible to A. L = the vocabulary of B, L L that of A. Suppose for all second order L-sentences φ: B = φ A = φ. Can write a second order L-sentence Θ(x, y) such that for all L -formulas φ(x) and any n N B = Θ( φ, n) A = φ(n), thus A = φ(n) A = Θ( φ, n). Let ψ(x) be the L -formula which says Θ(a, a) for the natural number" a, in the formal sense, that is the value of x. Let k = ψ(x). Now κ = Θ(k, k) A = ψ(k) A = (Θ(k, k)), a contradiction.

16 In summary: Validity of sentences of even low level" second order logic cannot be analyzed in terms of truth in a particular second order characterizable structure. In set theory: There is a Σ n -truth definition for Σ n -formulas. There are arbitrarily large α such that V α n V. So for a Σ n -sentence φ : φ is true V α = φ.

17 Outline Second order characterizable structures 1 Second order characterizable structures 2 Significance of the

18 M = (M, R 1,..., R n, f 1,..., f m, c 1,..., c k ), M countable. Does the second order theory of M determine M up to isomorphism? Ajtai showed in 1979 that this cannot be decided on the basis of CA (or ZFC) alone. We show: The non-categoricity phenomenon is quite general.

19 M = (M, R 1,..., R n, f 1,..., f m, c 1,..., c k ), M countable. Does the second order theory of M determine M up to isomorphism? Ajtai showed in 1979 that this cannot be decided on the basis of CA (or ZFC) alone. We show: The non-categoricity phenomenon is quite general.

20 M = (M, R 1,..., R n, f 1,..., f m, c 1,..., c k ), M countable. Does the second order theory of M determine M up to isomorphism? Ajtai showed in 1979 that this cannot be decided on the basis of CA (or ZFC) alone. We show: The non-categoricity phenomenon is quite general.

21 M = (M, R 1,..., R n, f 1,..., f m, c 1,..., c k ), M countable. Does the second order theory of M determine M up to isomorphism? Ajtai showed in 1979 that this cannot be decided on the basis of CA (or ZFC) alone. We show: The non-categoricity phenomenon is quite general.

22 Ajtai s First Theorem (1979) Theorem If V = L (or there is a second order definable well-order of R), then every countable model in a finite vocabulary is second order characterizable by a theory. A second order definable well-order of R means a well-order of P(N) such that for some second order formula φ(<, P, Q) for all A N, B N: A B (N, <, A, B) = φ(<, P, Q)}. If V = L, then there is a Σ 1 2 well-order of R.

23 Ajtai s First Theorem (1979) Proof. Assume V = L via φ(<, P, Q). Suppose M and M are second order equivalent. W.l.o.g. M = M = N and M is minimal in (mod coding) among models = M, same with M. We show M = M. Suppose S is in the vocabulary of M and M = S(a 1,..., a n ). M satisfies the sentence For some linear order < of order-type ω, the -minimal structure isomorphic to me satisfies S(a 1,..., a n )". Hence M has a linear order < of order-type ω such that the -minimal structure isomorphic to M satisfies S(a 1,..., a n ). Extend π : (M, < ) = (N, <) to π : (M, < ) = (M, <). By minimality, M = M. Then M = S(a 1,..., a n ) follows.

24 Ajtai s First Theorem (1979) Proof. Assume V = L via φ(<, P, Q). Suppose M and M are second order equivalent. W.l.o.g. M = M = N and M is minimal in (mod coding) among models = M, same with M. We show M = M. Suppose S is in the vocabulary of M and M = S(a 1,..., a n ). M satisfies the sentence For some linear order < of order-type ω, the -minimal structure isomorphic to me satisfies S(a 1,..., a n )". Hence M has a linear order < of order-type ω such that the -minimal structure isomorphic to M satisfies S(a 1,..., a n ). Extend π : (M, < ) = (N, <) to π : (M, < ) = (M, <). By minimality, M = M. Then M = S(a 1,..., a n ) follows.

25 Ajtai s Second Theorem Theorem (Ajtai 1979) It is consistent, relative to the consistency of ZF, that there is a countable model in a finite vocabulary which is not second order characterizable by a theory.

26 Ajtai s Second Theorem 1979 Proof. Let P be Cohen forcing for adding a generic set G N. Let F G be the set of A N with A G finite. We show that M = (ω F G, ω, <, E), where nex n X, is not characterizable by a second order theory. Let M = (ω F G, ω, <, E). Since M M, it suffices to prove that M and M are second order equivalent. In fact more is true: If Φ(x) is any formula of set theory, then Φ(M) Φ(M ). Suppose p Φ(Ṁ). Let G be a generic set extending p. Then V [G] = Φ(M). Let G be another generic, which agrees with G on p but is the complement of G elsewhere. Clearly M G = M G. Thus V [G ] = Φ(M ). But V [G] = V [G ], so V [G] = Φ(M ).

27 Ajtai s Second Theorem 1979 Proof. Let P be Cohen forcing for adding a generic set G N. Let F G be the set of A N with A G finite. We show that M = (ω F G, ω, <, E), where nex n X, is not characterizable by a second order theory. Let M = (ω F G, ω, <, E). Since M M, it suffices to prove that M and M are second order equivalent. In fact more is true: If Φ(x) is any formula of set theory, then Φ(M) Φ(M ). Suppose p Φ(Ṁ). Let G be a generic set extending p. Then V [G] = Φ(M). Let G be another generic, which agrees with G on p but is the complement of G elsewhere. Clearly M G = M G. Thus V [G ] = Φ(M ). But V [G] = V [G ], so V [G] = Φ(M ).

28 Outline Second order characterizable structures 1 Second order characterizable structures 2 Significance of the

29 Non-categoricity in uncountable models Lauri Keskinen, Amsterdam 2011: L 2 κω: Add to second order logic infinite conjunctions and disjunctions of length < κ. Ajtai(κ) says the L 2 κω-theory of any model of cardinality κ in a finite vocabulary determines the model up to isomorphism among models of cardinality κ" i.e. L 2 κω-equivalent models of cardinality κ in a finite vocabulary are isomorphic."

30 Uncountable models Lauri Keskinen, Amsterdam 2011: V = L implies Ajtai(κ) for all κ. For any κ there is a forcing extension in which κ is preserved and Ajtai(κ) fails. For any finite set of regular cardinals, Ajtai(κ) can hold in the set and fail for regular cardinals outside. Ajtai(ω) is consistent with n Woodin cardinals. Third order Ajtai(ω) is consistent even with a supercompact cardinal. A proper class of Woodin cardinals implies that Ajtai(ω) fails.

31 Hyttinen-Kangas-V. 2011: Extension of to models of first order theories, using stability theory and Shelah s Classification Theory.

32

33 Outline Second order characterizable structures Significance of the 1 Second order characterizable structures 2 Significance of the

34 Henkin structures Significance of the Definition A Henkin model is a pair (M, G), where M is a structure and G is a collection of relations on M, i.e. G n<ω P(Mn ). Full, if = instead of.

35 Truth definition for Henkin structures Significance of the M = s X m n φ M = s(p/x m n ) φ for some P P(M m ) (M, G) = s X m n φ (M, G) = s(p/x m n ) φ for some P P(M m ) G

36 The rationale behind Significance of the Some results about full models hold even for Henkin models. Some results can only be obtained for. Some results obtain for under weaker assumptions than for full models. Truth in can be axiomatized.

37 Significance of the The are assumed to satisfy all instances of the Comprehension Axioms: or equivalently, φ(ψ( z)/y ) Y φ, Y x 1... x m (Yx 1...x m ψ) for any second order ψ not containing Y free.

38 Outline Second order characterizable structures Significance of the 1 Second order characterizable structures 2 Significance of the

39 Significance of the There are obvious axioms and rules for second order logic, introduced by Hilbert-Ackermann 1928, among them the Comprehension Axioms and Axioms of Choice. Theorem (Henkin 1951) If T is consistent (does not prove a contradiction), then T has a Henkin model.

40 Significance of the Sketch. One describes a winning strategy of the second player" in a game, called the Model Existence Game (see Models and Games monograph CAP 2011). The strategy of II is to play so that at each point the set of played sentences is consistent with T (i.e. does not prove a contradiction). The winning strategy, when played against the best strategy of the first player", yields a Henkin model of T.

41 Outline Second order characterizable structures Significance of the 1 Second order characterizable structures 2 Significance of the

42 Significance of the are everywhere where there is consistency. This does not tell us so much about the mathematical objects we are usually interested in, as about the nature of consistency. Some properties of the Henkin model cannot be expressed: countable, uncountable, finite, infinite. Henkin models are a tool, and this tool is very flexible. The axiom P 2 has non-standard. Whatever we prove about the standard model will be true in the non-standard models, too. This is what provability means.

43 Significance of the are everywhere where there is consistency. This does not tell us so much about the mathematical objects we are usually interested in, as about the nature of consistency. Some properties of the Henkin model cannot be expressed: countable, uncountable, finite, infinite. Henkin models are a tool, and this tool is very flexible. The axiom P 2 has non-standard. Whatever we prove about the standard model will be true in the non-standard models, too. This is what provability means.

44 Significance of the are everywhere where there is consistency. This does not tell us so much about the mathematical objects we are usually interested in, as about the nature of consistency. Some properties of the Henkin model cannot be expressed: countable, uncountable, finite, infinite. Henkin models are a tool, and this tool is very flexible. The axiom P 2 has non-standard. Whatever we prove about the standard model will be true in the non-standard models, too. This is what provability means.

45 Significance of the To keep in mind about : Second order sentences are not full or Henkin, only models are. The sentences of second order logic do not know" what semantics is being used.

46 Significance of the To keep in mind about : Second order sentences are not full or Henkin, only models are. The sentences of second order logic do not know" what semantics is being used.

47 Significance of the To keep in mind about : Second order sentences are not full or Henkin, only models are. The sentences of second order logic do not know" what semantics is being used.

48 The usefulness of : Significance of the To give a (finite) proof of a second order φ from a second order theory T one can use the semantical method: Suppose (M, G) = T. We show (M, G) = φ. We have proved not only that φ is true in the intended model of T (if there was one), but that φ is true in the entire cloud inside which the intended model of T is lurking. One can hardly find a convincing argument for φ logically following from T unless there was an actual formal proof of φ from T and the Comprehension (and Choice) axioms.

49 The usefulness of : Significance of the To give a (finite) proof of a second order φ from a second order theory T one can use the semantical method: Suppose (M, G) = T. We show (M, G) = φ. We have proved not only that φ is true in the intended model of T (if there was one), but that φ is true in the entire cloud inside which the intended model of T is lurking. One can hardly find a convincing argument for φ logically following from T unless there was an actual formal proof of φ from T and the Comprehension (and Choice) axioms.

50 The usefulness of : Significance of the To give a (finite) proof of a second order φ from a second order theory T one can use the semantical method: Suppose (M, G) = T. We show (M, G) = φ. We have proved not only that φ is true in the intended model of T (if there was one), but that φ is true in the entire cloud inside which the intended model of T is lurking. One can hardly find a convincing argument for φ logically following from T unless there was an actual formal proof of φ from T and the Comprehension (and Choice) axioms.

51 Significance of the The transfer from M(M = T M = φ) to M, G((M, G) = T (M, G) = φ) is in practice insignificant, but it has important philosophical content. Paying attention to G in the proof gives a reduction from an infinitistic inference to a finitist one.