Reflection ranks and proof theoretic ordinals (based on joint work - PowerPoint PPT Presentation

Reflection ranks and proof theoretic ordinals (based on joint work with James Walsh) Fedor Pakhomov Steklov Mathematical Institute, Moscow pakhf@mi.ras.ru Logic Colloquium 2018, Udine 24 July 2018

≺ Con -Order def ⇒ U proves consistence of T . T ≺ Con U ⇐ Empirical fact: ≺ Con is a linear well-founded preorder on natural theories I Σ 1 ≺ Con . . . ≺ Con I Σ n ≺ Con PA ≡ Con ACA 0 ACA 0 ≺ Con Π 1 1 -CA 0 ≺ Con Π 1 2 -CA 0 ≺ Con . . . ≺ Con Π 1 ∞ -CA 0 = PA 2 PA 2 ≺ Con PA 3 ≺ Con . . . ≺ Con PA ∞ ≡ Con Z Z ≺ Con Z +∆ 0 -Coll ≺ Con Z +Π 1 -Coll ≺ Con . . . ≺ Con Z +Π ∞ -Coll = ZF ZF ≺ Con ZFC + ∃ κ κ is inaccessible ≺ Con . . . Although it is possible to construct artificial examples of descending chains consisting of true theories. T 0 ≻ Con T 1 ≻ Con T 2 ≻ Con . . .

Π 1 1 soundndess and Π 1 1 reflection Let T be an r.e. extension of ACA 0 . ACA 0 = PA + second order axiom of induction + ∃ X ∀ x ( ϕ ( n ) ↔ x ∈ X ) , for all arithmetical ( Π 0 ∞ ) formulas ϕ ( x ) . The Π 1 1 ( T ) is Π 1 1 reflection principle RFN Π 1 1 sentence expressing T is Π 1 1 -sound, e.g. T proves only true Π 1 1 sentences. More formally RFN Π 1 1 ( T ) is given by the sentence ∀ ϕ ∈ Π 1 1 ( Prv ( T , ϕ ) → Tr Π 1 1 ( ϕ )) , 1 ( x ) is the partial truth definition for Π 1 where Tr Π 1 1 formulas.

Well-foundedness in reflection order We put def T ≺ Π 1 ⇐ ⇒ U ⊢ RFN Π 1 1 ( T ) . 1 U Note that T ≺ Π 1 1 U ⇒ T ≺ Con U . Theorem 1 on Π 1 The restriction of ≺ Π 1 1 -sound extensions of ACA 0 is a well-founded relation.

Proof of Well-Foundedness of ≺ Π 1 1 The negation of our theorem is the sentence DS 1 starting with Π 1 DS: “there is a descending chain in ≺ Π 1 1 -sound r.e. theory” We will show that ACA 0 + DS ⊢ Con ( ACA 0 + DS ) . Then by G¨ odel’s second incompleteness theorem ACA 0 + DS is inconsistent and hence ACA 0 ⊢ ¬ DS. Let us reason in ACA 0 + DS. We have sequence T 0 ≻ Π 1 1 T 1 ≻ Π 1 1 . . . , where T 0 is Π 1 1 -sound. Let S be the Σ 1 1 -sentence saying that “there is a descending sequence in ≺ Π 1 1 starting from T 1 .” Since S is true and T 0 is Π 1 1 -sound, there is a (countably coded) model M | = T 0 + S But since T 0 proves Π 1 1 -soundness of T 1 , M | = DS .

The case of RCA 0 Over RCA 0 there are no truth definition for the class Π 1 1 but there are truth definitions for smaller classes Π 1 1 (Π 0 n ) , e.g. formulas of the form ∀ � X ϕ , where ϕ ∈ Π 0 n . And we have reflection principles RFN Π 1 n ) ( T ) . 1 (Π 0 Theorem 3 ) on Π 1 1 (Π 0 The restriction of ≺ Π 1 3 ) -sound extensions of RCA 0 is 1 (Π 0 a well-founded relation. Clarification : Note that we need partial truth definition for class of formulas Γ to make reflection principle RFN Γ a single sentence. Otherwise we put RFN Γ be the scheme ∀ � x ( Prv ( T , ϕ ( � x )) → ϕ ( � x )) , where ϕ ∈ Γ .

Reflection in first-order arithmetic Over the system of first-order arithmetic EA we have partial truth definitions Tr Π 0 n ( x ) and reflection principles RFN Π 0 n ( T ) . Theorem (Friedman, Smorynski, Solovay) There are no recursive sequences of theories � T i | i ∈ N � such that T 0 is consistent and EA ⊢ ∀ x Prv ( T x , � Con ( T x + 1 ) � ) . Theorem There are no recursive sequences of theories � T i | i ∈ N � such that T 0 is Π 0 3 -sound and T 0 ≻ Π 0 3 T 1 ≻ Π 0 3 . . .

Recursive descending chains Recursive descending chain in ≺ Π 0 2 : T 0 ≻ Π 0 2 T 1 ≻ Π 0 2 T 2 ≻ Π 0 2 . . . 2 ( PA ) or RFN p − n T n : I Σ 1 + “ either RFN Π 0 2 ( I Σ 1 ) , where p is G¨ odel Π 0 number of the first proof of false Σ 0 1 sentence in PA” Note that all T n are true arithmeical theories.

Reflection Rank For an r.e. extension T of ACA 0 we put | T | ACA 0 = α if T is in well-founded part of ≺ Π 1 1 and α is it’s well-founded rank | T | ACA 0 = ∞ , otherwise More standard measure is Π 1 1 proof-theoretic ordinal: | T | WO = sup {| α | | α is recursive linear order and T ⊢ WO ( α ) } . Reflection ranks and proof-theoretic ordinals of some theories: | · | ACA 0 | · | WO ACA 0 0 ε 0 ACA 0 + Con ( ACA 0 ) 0 ε 0 ACA 0 + RFN Π 1 1 ( ACA 0 ) 1 ε 1 ACA ′ ω ε ω 0 ACA ε 0 ε ε 0 ACA + ϕ ( 2 , 0 ) ϕ ( 2 , 0 ) 0 ATR 0 Γ 0 Γ 0

Iterations of reflection principles For recursive ordinal notations α we could define iterations RFN α Γ ( T ) : ◮ RFN 0 Γ ( T ) = T ◮ RFN α + 1 ( T ) = T + RFN Γ ( RFN α Γ ( T )) Γ ◮ RFN λ RFN α Γ ( T ) = � Γ ( T ) , λ ∈ Lim . α<λ Theorem (Turing) For each true Π 1 sentence F there is recursive ordinal notation α Con α ( PA ) ⊢ F . Theorem (Feferman) For each true Π 0 ∞ sentence F there is recursive ordinal notation α RFN α ∞ ( PA ) ⊢ F . Π 0

Iterations of Π 1 1 -reflection Theorem RFN α 3 ) RFN ε α 1 ( ACA 0 ) ≡ Π 1 3 ) ( RCA 0 ) 1 (Π 0 Π 1 Π 1 1 (Π 0 Proposition | RFN β 3 ) ( RCA 0 ) | RCA 0 = | β | Π 1 1 (Π 0 Proposition ACA 0 ⊢ ∀ α ( WO ( α ) ↔ RFN α + 1 3 ) ( RCA 0 )) Π 1 1 (Π 0 Corollary | RFN α 1 ( ACA 0 ) | WO = | ε α | . Π 1

Proving RFN α 3 ) RFN ε α 1 ( ACA 0 ) ≡ Π 1 3 ) ( RCA 0 ) 1 (Π 0 Π 1 Π 1 1 (Π 0 Let us consider pseudo- Π 1 1 language Π 0 ∞ , i.e. arithmetical formulas ϕ ( X ) with free unary predicate X . We have embedding of pseudo- Π 1 1 language into second-order arithmetic ϕ ( X ) �− → ∀ X ϕ ( X ) . RFN α ∞ RFN α 1 ( ACA 0 ) ≡ Π 0 ∞ ( PA ( X )) , Π 1 Π 0 RFN α 3 RFN α 3 ) ( RCA 0 ) ≡ Π 0 3 ( I Σ 1 ( X )) . Π 1 1 (Π 0 Π 0 Schmerl-style formula for uniform pseudo- Π 1 1 reflection RFN α 3 RFN ε α ∞ ( PA ( X )) ≡ Π 0 3 ( I Σ 1 ) Π 0 Π 0 Thus RFN α 1 ( ACA 0 ) ≡ Π 0 ∞ RFN α ∞ ( PA ( X )) ≡ Π 0 3 RFN ε α 3 ( I Σ 1 ) ≡ Π 0 3 RFN ε α 3 ) ( RCA 0 ) Π 1 Π 0 Π 0 Π 1 1 (Π 0

Calculus RC 0 Beklemishev approach to proof of Schmerl formula employs ordinal notation system based on reflection principles. Reflection calculus RC: Formulas: F ::= ⊤ | F ∧ F | ✸ n F , where n ranges over N . Sequents: A ⊢ B , for RC-formulas A and B . 1. A ⊢ A ; A ⊢ ⊤ ; if A ⊢ B and B ⊢ C then A ⊢ C ; 2. A ∧ B ⊢ A ; A ∧ B ⊢ B ; if A ⊢ B and A ⊢ C then A ⊢ B ∧ C ; 3. if A ⊢ B then ✸ n A ⊢ ✸ n B , for all n ∈ N ; 4. ✸ n ✸ n A ⊢ ✸ n A , for every n ∈ N ; 5. ✸ n A ⊢ ✸ m A , for all n > m ; 6. ✸ n A ∧ ✸ m B ⊢ ✸ n ( A ∧ ✸ m B ) , for all n > m .

Beklemishev’s Ordinal Notation System def A < 0 B ⇐ ⇒ B ⊢ ✸ 0 A def A ∼ B ⇐ ⇒ A ⊢ B and B ⊢ A Theorem (Beklemishev) ( RC 0 / ∼ , < 0 ) is a well-ordering with order type ε 0 . It were done by Beklemishev by embedding this system in Cantor ordinal notation system for ε 0 .

Well-Foundedness Proof Let us interpret RC-formulas by L 2 -theories. We interpret ⊤ as ⊤ ⋆ = ACA 0 . And we interpret ✸ n A as ( ✸ n A ) ⋆ = RFN Π 1 n + 1 ( A ⋆ ) . It is easy to see that A ⊢ B implies A ⋆ ⊢ B ⋆ . Hence A < 0 B implies A ⋆ < Π 1 1 B ⋆ . Thus < 0 is a well-founded relation on the set of RC 0 formulas.

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