Generic cuts in models of Peano arithmetic

Transcription

1 Generic cuts in models of Peano arithmetic Tin Lok Wong University of Birmingham, United Kingdom Joint work with Richard Kaye (Birmingham) 5 August, 2009

2 Preliminary definitions L A is the first-order language for arithmetic {0, 1, +,, <}.

3 Preliminary definitions L A is the first-order language for arithmetic {0, 1, +,, <}. Peano Arithmetic (PA) is the L A -theory consisting of axioms for the non-negative part of discretely ordered rings

4 Preliminary definitions L A is the first-order language for arithmetic {0, 1, +,, <}. Peano Arithmetic (PA) is the L A -theory consisting of axioms for the non-negative part of discretely ordered rings and the induction axiom z [ ϕ(0, z) x ( ϕ(x, z) ϕ(x + 1, z) ) xϕ(x, z) ]. for each L A -formula ϕ(x, z).

5 Aim Understand structures of the form (M, I) where M = PA and I is cut of M. {{ M {{ I

6 Aim Understand structures of the form (M, I) where M = PA and I is cut of M. {{ M {{ I How complicated is Th(M, I) in relation to Th(M)?

7 Aim Understand structures of the form (M, I) {{ M {{ I where M = PA and I is cut of M. How complicated is Th(M, I) in relation to Th(M)? How does Aut(M, I) sit inside Aut(M)?

8 Aim Understand structures of the form (M, I) {{ M {{ I where M = PA and I is cut of M. How complicated is Th(M, I) in relation to Th(M)? How does Aut(M, I) sit inside Aut(M)? Is (M, I) easier to study than (I, SSy I (M)) where SSy I (M) = {X I : X M is definable with parameters}?

9 Arithmetic saturation

10 Arithmetic saturation Definition A model M of PA is recursively saturated if every recursive type over M is realized in M.

11 Arithmetic saturation Definition A model M of PA is recursively saturated if every recursive type over M is realized in M. Every formula is a number via a Gödel numbering.

12 Arithmetic saturation Definition A model M of PA is recursively saturated if every recursive type over M is realized in M. Fact Countable recursively saturated models of PA are ω-homogeneous.

13 Arithmetic saturation Definition A model M of PA is recursively saturated if every recursive type over M is realized in M. Fact Countable recursively saturated models of PA are ω-homogeneous. if two elements satisfy the same formulas, then there is an automorphism bringing one to the other.

14 Arithmetic saturation Definition A model M of PA is recursively saturated if every recursive type over M is realized in M. Fact Countable recursively saturated models of PA are ω-homogeneous. Definition A model M of PA is arithmetically saturated if it is recursively saturated and (N, SSy N (M)) = ACA 0.

15 Topological background Fix a countable arithmetically saturated model M of PA.

16 Topological background Fix a countable arithmetically saturated model M of PA. points

17 Topological background Fix a countable arithmetically saturated model M of PA. Definition A cut of M is elementary if it is an elementary substructure of M. We write I e M for I is an elementary cut of M. points

18 Topological background Fix a countable arithmetically saturated model M of PA. Definition A cut of M is elementary if it is an elementary substructure of M. We write I e M for I is an elementary cut of M. points open sets

19 Topological background Fix a countable arithmetically saturated model M of PA. Definition A cut of M is elementary if it is an elementary substructure of M. We write I e M for I is an elementary cut of M. points Definition An elementary interval is a nonempty set of the form where a, b M. [[a, b]] = {I e M : a I < b} open sets

20 Topological background Fix a countable arithmetically saturated model M of PA. Definition A cut of M is elementary if it is an elementary substructure of M. We write I e M for I is an elementary cut of M. points Definition An elementary interval is a nonempty set of the form where a, b M. [[a, b]] = {I e M : a I < b} open sets Fact The elementary intervals generate a topology on the collection of all elementary cuts.

21 Topological background Fix a countable arithmetically saturated model M of PA. Definition A cut of M is elementary if it is an elementary substructure of M. We write I e M for I is an elementary cut of M. points Definition An elementary interval is a nonempty set of the form where a, b M. [[a, b]] = {I e M : a I < b} open sets Fact The space of elementary cuts is homeomorphic to the Cantor set.

22 Genericity Definition A subset of a topological space is comeagre if it contains a countable intersection of dense open sets.

23 Genericity Comeagre means large. Definition A subset of a topological space is comeagre if it contains a countable intersection of dense open sets.

24 Genericity Comeagre means large. A property is generic if it is satisfied by a large number of cuts. Definition A subset of a topological space is comeagre if it contains a countable intersection of dense open sets.

25 Genericity Comeagre means large. A property is generic if it is satisfied by a large number of cuts. Definition A subset of a topological space is comeagre if it contains a countable intersection of dense open sets. Definition An elementary cut is generic if it is contained in any comeagre set of elementary cuts that is closed under the automorphisms of M.

26 Genericity Comeagre means large. A property is generic if it is satisfied by a large number of cuts. Definition A subset of a topological space is comeagre if it contains a countable intersection of dense open sets. Definition An elementary cut is generic if it is contained in any comeagre set of elementary cuts that is closed under the automorphisms of M. A generic cut satisfies all generic properties.

27 Pregeneric intervals Theorem Let c M and [[a, b]] be an elementary interval. Then there is an elementary subinterval [[r, s]] of [[a, b]] such that for every elementary subinterval [[u, v]] of [[r, s]] there is an elementary subinterval [[r, s ]] of [[u, v]] such that (M, r, s, c) = (M, r, s, c).

28 Pregeneric intervals Theorem Let c M and [[a, b]] be an elementary interval. Then there is an elementary subinterval [[r, s]] of [[a, b]] such that for every elementary subinterval [[u, v]] of [[r, s]] there is an elementary subinterval [[r, s ]] of [[u, v]] such that (M, r, s, c) = (M, r, s, c). This subinterval [[r, s]] is said to be pregeneric over c.

29 Pregeneric intervals Theorem Let c M and [[a, b]] be an elementary interval. Then there is an elementary subinterval [[r, s]] of [[a, b]] such that for every elementary subinterval [[u, v]] of [[r, s]] there is an elementary subinterval [[r, s ]] of [[u, v]] such that (M, r, s, c) = (M, r, s, c). This subinterval [[r, s]] is said to be pregeneric over c. self-similarity

30 Pregeneric intervals Theorem Let c M and [[a, b]] be an elementary interval. Then there is an elementary subinterval [[r, s]] of [[a, b]] such that for every elementary subinterval [[u, v]] of [[r, s]] there is an elementary subinterval [[r, s ]] of [[u, v]] such that (M, r, s, c) = (M, r, s, c). This subinterval [[r, s]] is said to be pregeneric over c. Proof. A tree argument.

31 Generic cuts Take an enumeration (c n ) n N of M.

32 Generic cuts Take an enumeration (c n ) n N of M. Starting with an arbitrary elementary interval [[a 0, b 0 ]], construct a sequence [[a 0, b 0 ]] [[a 1, b 1 ]] [[a 2, b 2 ]] such that [[a n+1, b n+1 ]] is pregeneric over c n for all n N.

33 Generic cuts Take an enumeration (c n ) n N of M. Starting with an arbitrary elementary interval [[a 0, b 0 ]], construct a sequence [[a 0, b 0 ]] [[a 1, b 1 ]] [[a 2, b 2 ]] such that [[a n+1, b n+1 ]] is pregeneric over c n for all n N. Then there is a unique elementary cut in n N [[a n, b n ]].

34 Generic cuts Take an enumeration (c n ) n N of M. Starting with an arbitrary elementary interval [[a 0, b 0 ]], construct a sequence [[a 0, b 0 ]] [[a 1, b 1 ]] [[a 2, b 2 ]] such that [[a n+1, b n+1 ]] is pregeneric over c n for all n N. Then there is a unique elementary cut in n N [[a n, b n ]]. Theorem The cuts constructed in this way are exactly the generic cuts.

35 Generic cuts Take an enumeration (c n ) n N of M. Starting with an arbitrary elementary interval [[a 0, b 0 ]], construct a sequence [[a 0, b 0 ]] [[a 1, b 1 ]] [[a 2, b 2 ]] such that [[a n+1, b n+1 ]] is pregeneric over c n for all n N. Then there is a unique elementary cut in n N [[a n, b n ]]. Theorem The cuts constructed in this way are exactly the generic cuts. Proof. Back-and-forth.

36 Generic cuts under automorphisms Proposition (M, I 1 ) = (M, I 2 ) for all generic cuts I 1, I 2 in M.

37 Generic cuts under automorphisms Proposition (M, I 1 ) = (M, I 2 ) for all generic cuts I 1, I 2 in M. Theorem If I is a generic cut of M and c, d I such that tp(c) = tp(d), then (M, I, c) = (M, I, d).

38 Description of truth Theorem Let I be a generic cut of M. Then for all c, d M, if and only if tp(c) = tp(d), and (M, I, c) = (M, I, d) for every L A -formula ϕ(x, z), {x I : M = ϕ(x, c)} has an upper bound in I {x I : M = ϕ(x, d)} has an upper bound in I.

39 Description of truth Theorem Let I be a generic cut of M. Then for all c, d M, Quantifier elimination? (M, I, c) = (M, I, d) if and only if tp(c) = tp(d), and for every L A -formula ϕ(x, z), {x I : M = ϕ(x, c)} has an upper bound in I {x I : M = ϕ(x, d)} has an upper bound in I.

40 Conclusion What we did

41 Conclusion What we did Picked out a more tractable (M, I) for each countable arithmetically saturated model M.

42 Conclusion What we did Picked out a more tractable (M, I) for each countable arithmetically saturated model M. Obtained some information about the automorphisms of this (M, I).

43 Conclusion What we did Picked out a more tractable (M, I) for each countable arithmetically saturated model M. Obtained some information about the automorphisms of this (M, I). Understood more about the fine structure of countable arithmetically saturated models.

44 Conclusion What we did Picked out a more tractable (M, I) for each countable arithmetically saturated model M. Obtained some information about the automorphisms of this (M, I). Understood more about the fine structure of countable arithmetically saturated models. What next? Let I be a generic cut.

45 Conclusion What we did Picked out a more tractable (M, I) for each countable arithmetically saturated model M. Obtained some information about the automorphisms of this (M, I). Understood more about the fine structure of countable arithmetically saturated models. What next? Let I be a generic cut. What is special about (I, SSy I (M)) and Th(M, I)?

46 Conclusion What we did Picked out a more tractable (M, I) for each countable arithmetically saturated model M. Obtained some information about the automorphisms of this (M, I). Understood more about the fine structure of countable arithmetically saturated models. What next? Let I be a generic cut. What is special about (I, SSy I (M)) and Th(M, I)? How does Aut(M, I) sit inside Aut(M)?

47 Conclusion What we did Picked out a more tractable (M, I) for each countable arithmetically saturated model M. Obtained some information about the automorphisms of this (M, I). Understood more about the fine structure of countable arithmetically saturated models. What next? Let I be a generic cut. What is special about (I, SSy I (M)) and Th(M, I)? How does Aut(M, I) sit inside Aut(M)? Investigate the existential closure properties of (M, I).