Models of Weak K onigs Lemma Tin Lok Wong Kurt G odel Research - PowerPoint PPT Presentation

Models of Weak K¨ onig’s Lemma Tin Lok Wong Kurt G¨ odel Research Center for Mathematical Logic Vienna, Austria Joint work with Ali Enayat (Gothenburg) 10 September, 2015 Financial support from FWF Project P24654-N25 is acknowledged.

This talk Weak K¨ onig’s Lemma (WKL) Every infinite 0–1 tree has an infinite branch. Plan 1. Motivation models of WKL ≈ coded subsets in 2. Self-embeddings end extensions 3. Set-extensions 4. Conclusion

IΣ 0 = | M First-order arithmetic n ∈ N ◮ L I = { 0 , 1 , + , × , < } . ◮ A quantifier is bounded if it is of the form ∀ v < t or ∃ v < t . N ◮ An L I -formula is ∆ 0 if all its quantifiers are bounded. . . . ◮ Σ n = {∃ ¯ v 1 ∀ ¯ v 2 · · · Q ¯ v n θ (¯ v , ¯ x ) : θ ∈ ∆ 0 } . 1 + 1 ◮ The dual is called Π n . 1 ◮ A formula is ∆ n if it is both Σ n and Π n . 0 ◮ IΣ n consists of some basic axioms (PA − ) and for every θ ∈ Σ n , � � θ (0) ∧ ∀ x θ ( x ) → θ ( x + 1) → ∀ x θ ( x ) . ◮ BΣ n +1 consists of the axioms of IΣ 0 and for every θ ∈ Σ n +1 , � � ∀ a ∀ x < a ∃ y θ ( x , y ) → ∃ b ∀ x < a ∃ y < b θ ( x , y ) . ◮ exp asserts the totality of x �→ 2 x . Theorem (Paris–Kirby 1978; Parsons 1970; Parikh 1971) IΣ n +1 ⊢ BΣ n +1 ⊢ IΣ n for all n ∈ N ; and IΣ 1 ⊢ exp but BΣ 1 � exp.

IΣ n = | M Cuts and end extensions n ∈ N ⇓ Definition Let ■ , M | = IΣ 0 . Say ■ is a cut of M , or Σ n -Def( M ) �∋ N M is an end extension of ■ , if ■ ⊆ M and . . . ∀ i ∈ ∀ m ∈ M \ i � m . ■ ■ 1 + 1 1 In this case, write ■ ⊆ e M . 0 ◮ IΣ n consists of some basic axioms (PA − ) and for every θ ∈ Σ n , � � θ (0) ∧ ∀ x θ ( x ) → θ ( x + 1) → ∀ x θ ( x ) . Proposition (folklore) (1) N is a cut of every model of IΣ 0 , called the standard cut . (2) If M �∼ = N and M | = IΣ n , then N is not Σ n -definable in M . � �� � saturation condition M is nonstandard

Second-order arithmetic ◮ L I I = { 0 , 1 , + , × , <, ∈} has a number sort and a set sort . ◮ A quantifier is bounded if it is of the form ∀ v < t or ∃ v < t . ◮ ∆ 0 0 , Σ 0 n , Π 0 n , ∆ 0 n are defined as in L I . ◮ Formulas in � n ∈ N Σ 0 n are called arithmetical . ◮ ∆ 0 1 - CA stands for the ∆ 0 1 -comprehension axiom. ◮ RCA 0 = IΣ 0 1 + ∆ 0 0 = BΣ 0 1 + exp + ∆ 0 RCA ∗ 1 -CA. 1 -CA. ◮ WKL 0 = RCA 0 + WKL. WKL ∗ 0 = RCA ∗ 0 + WKL. ◮ If M | = IΣ 1 , then ( M , ∆ 1 -Def( M )) | = RCA 0 + ¬ WKL. ◮ If M | = BΣ 1 + exp, then ( M , ∆ 1 -Def( M )) | = RCA ∗ 0 + ¬ WKL. = PA = � ◮ If M | n ∈ N IΣ n , then ( M , Def( M )) | = WKL 0 . Theorem (Harrington 1977) If σ = ∀ X ϕ ( X ) where ϕ is arithmetical, then WKL 0 ⊢ σ ⇒ RCA 0 ⊢ σ.

Coded sets Let M ⊆ e K | = IΣ 0 . ◮ Say c ∈ K codes S ⊆ M if S = { x ∈ M : the x th prime divides c } . ◮ Denote by Cod( K / M ) the set of all S ⊆ M coded in K . Theorem (Scott 1962) = WKL ∗ If M � e K | = IΣ 0 and M | = exp, then ( M , Cod( K / M )) | 0 . Theorem (Enayat–W) = IΣ 0 The following are equivalent for a countable ( M , X ) | 0 + exp. (a) ( M , X ) | = WKL ∗ 0 . (b) X = Cod( K / M ) for some K � e M satisfying IΣ 0 .

Self-embeddings (pointwise fixing an initial segment) Theorem (H. Friedman 1973; Dimitracopoulos–Paris 1988) For every countable nonstandard M | = IΣ 1 , there exist ■ � e M and an isomorphism M → ■ . Theorem (Ressayre 1987) The following are equivalent for all countable M | = IΣ 0 . (a) M �∼ = N and M | = IΣ 1 . (b) For every a ∈ M , there exist ■ � e M and an isomorphism M → ■ which fixes all x < a . Theorem (Tanaka 1997) = IΣ 0 The following are equivalent for all countable ( M , X ) | 0 . (a) M �∼ = N and ( M , X ) | = WKL 0 . (b) For every a ∈ M , there exist ■ � e M and an isomorphism ( M , X ) → ( ■ , Cod( M / ■ )) which fixes all x < a .

Self-embeddings closure of Σ 1 under Boolean operations and bounded quantification Proposition (folklore) If M �∼ = N and M | = IΣ 1 , then N is not ∆ 0 (Σ 1 )-definable in M . Theorem (Dimitracopoulos–Paris 1988) The following are equivalent for a countable M | = IΣ 0 + exp. (a) M ∼ = ■ for some ■ � e M . (b) M | = BΣ 1 and N is not parameter-free ∆ 0 (Σ 1 )-definable in M . Theorem (Enayat–W) = IΣ 0 The following are equivalent for a countable ( M , X ) | 0 + exp. (a) ( M , X ) ∼ = ( ■ , Cod( M / ■ )) for some ■ � e M . = WKL ∗ (b) ( M , X ) | 0 and N is not parameter-free ∆ 0 (Σ 1 )-definable in M . not related to X

年 巻 数理解析研究所講究録 Tanaka’s Conjecture not true for σ = ∃ X ϕ ( X ) Theorem (Harrington 1977) in general If σ = ∀ X ϕ ( X ) where ϕ is arithmetical, then WKL 0 ⊢ σ ⇒ RCA 0 ⊢ σ. Tanaka’s Conjecture (1995) If σ = ∀ X ∃ ! Y ϕ ( X , Y ) where ϕ is arithmetical, then WKL 0 ⊢ σ ⇒ RCA 0 ⊢ σ.

The model theory behind Tanaka’s Conjecture Theorem (Simpson–Tanaka–Yamazaki 2002) If σ = ∀ X ∃ ! Y ϕ ( X , Y ) where ϕ is arithmetical, then WKL 0 ⊢ σ ⇒ RCA 0 ⊢ σ. Harrington: σ = ∀ X ϕ ( X ) Lemma (Harrington 1977) Every countable ( M , X ) | = RCA 0 can be extended to ( M , Y ) | = WKL 0 . Lemma (Simpson–Tanaka–Yamazaki 2002) Every countable ( M , X ) | = RCA 0 can be extended to ( M , Y 1 ) , ( M , Y 2 ) | = WKL 0 such that (a) Y 1 ∩ Y 2 = X ; and (b) ( M , Y 1 ) and ( M , Y 2 ) satisfy the same formulas with parameters from ( M , X ).

Models of WKL ≈ coded subsets in end extensions ◮ Ressayre, Tanaka: Having an isomorphism onto a proper cut fixing any given initial segment characterizes IΣ 1 and WKL 0 . ◮ Dimitracopoulos–Paris, Enayat–W: Having an isomorphism onto a proper cut is a sign of saturation. ◮ Simpson–Tanaka–Yamazaki: Any countable ( M , X ) | = RCA 0 can be extended to ( M , Y 1 ) , ( M , Y 2 ) | = WKL 0 with minimal intersection such that the same formulas with parameters from ( M , X ) are satisfied in them. Questions = RCA ∗ = WKL ∗ (1) Can every ( M , X ) | 0 be extended to ( M , Y ) | 0 ? (2) Scott 1962: Given ( M , X ) | = WKL 0 , can one always find K � e M satisfying IΣ 0 such that Cod( K / M ) = X ? = RCA ∗ (3) Can every countable ( M , X ) | 0 be extended to ( M , ∆ 0 1 -Def( M , A )) | = RCA ∗ 0 for some A ⊆ M ?