Prolongable Satisfaction Classes and Iterations of Uniform - PowerPoint PPT Presentation

Prolongable Satisfaction Classes and Iterations of Uniform Reflection over PA Mateusz Łełyk Institute of Philosophy, University of Warsaw Wormshop 2017 , Moscow, October 23, 2017

Very brief introduction This research is about the relations between partial inductive satisfaction classes, full satisfaction classes and iterated reflection principles. Mateusz Łełyk (IF UW) October 23, 2017, Moscow 2 / 17

Very brief introduction This research is about the relations between partial inductive satisfaction classes, full satisfaction classes and iterated reflection principles. Its primary motivation was entirely model-theoretical. Mateusz Łełyk (IF UW) October 23, 2017, Moscow 2 / 17

Very brief introduction This research is about the relations between partial inductive satisfaction classes, full satisfaction classes and iterated reflection principles. Its primary motivation was entirely model-theoretical. However it started paying back and after all I’m going to give Mateusz Łełyk (IF UW) October 23, 2017, Moscow 2 / 17

Very brief introduction This research is about the relations between partial inductive satisfaction classes, full satisfaction classes and iterated reflection principles. Its primary motivation was entirely model-theoretical. However it started paying back and after all I’m going to give a new proof of conservativity of CS 0 (the theory of a ∆ 0 -inductive full satisfaction class) over ω -times iterated uniform reflection over PA ( UR ω (PA)). Mateusz Łełyk (IF UW) October 23, 2017, Moscow 2 / 17

Very brief introduction This research is about the relations between partial inductive satisfaction classes, full satisfaction classes and iterated reflection principles. Its primary motivation was entirely model-theoretical. However it started paying back and after all I’m going to give a new proof of conservativity of CS 0 (the theory of a ∆ 0 -inductive full satisfaction class) over ω -times iterated uniform reflection over PA ( UR ω (PA)). a proof of conservativity of CS 0 + BΣ 1 ( S ) over UR ω (PA). Mateusz Łełyk (IF UW) October 23, 2017, Moscow 2 / 17

Very brief introduction This research is about the relations between partial inductive satisfaction classes, full satisfaction classes and iterated reflection principles. Its primary motivation was entirely model-theoretical. However it started paying back and after all I’m going to give a new proof of conservativity of CS 0 (the theory of a ∆ 0 -inductive full satisfaction class) over ω -times iterated uniform reflection over PA ( UR ω (PA)). a proof of conservativity of CS 0 + BΣ 1 ( S ) over UR ω (PA). The methods essentially consists in arithmetizing some well-known proofs. Mateusz Łełyk (IF UW) October 23, 2017, Moscow 2 / 17

Full satisfaction class Let M | = PA and S be a fresh binary predicate. A full satisfation class on M is a set S ⊆ M 2 such that the following L PA ∪ { S } sentences are true in ( M , S ): ( S ( s = t , α ) ≡ s α = t α ). 1 ∀ s , t ∀ α ∈ Asn( s , t ) Mateusz Łełyk (IF UW) October 23, 2017, Moscow 3 / 17

Full satisfaction class Let M | = PA and S be a fresh binary predicate. A full satisfation class on M is a set S ⊆ M 2 such that the following L PA ∪ { S } sentences are true in ( M , S ): ( S ( s = t , α ) ≡ s α = t α ). 1 ∀ s , t ∀ α ∈ Asn( s , t ) 2 ∀ φ ∀ α ∈ Asn( φ ) ( S ( ¬ φ, α ) ≡ ¬ S ( φ, α )). Mateusz Łełyk (IF UW) October 23, 2017, Moscow 3 / 17

Full satisfaction class Let M | = PA and S be a fresh binary predicate. A full satisfation class on M is a set S ⊆ M 2 such that the following L PA ∪ { S } sentences are true in ( M , S ): ( S ( s = t , α ) ≡ s α = t α ). 1 ∀ s , t ∀ α ∈ Asn( s , t ) 2 ∀ φ ∀ α ∈ Asn( φ ) ( S ( ¬ φ, α ) ≡ ¬ S ( φ, α )). 3 ∀ φ ∀ ψ ∀ α ∈ Asn( φ, ψ ) ( S ( φ ∨ ψ, α ) ≡ S ( φ, α ) ∨ S ( ψ, α )). Mateusz Łełyk (IF UW) October 23, 2017, Moscow 3 / 17

Full satisfaction class Let M | = PA and S be a fresh binary predicate. A full satisfation class on M is a set S ⊆ M 2 such that the following L PA ∪ { S } sentences are true in ( M , S ): ( S ( s = t , α ) ≡ s α = t α ). 1 ∀ s , t ∀ α ∈ Asn( s , t ) 2 ∀ φ ∀ α ∈ Asn( φ ) ( S ( ¬ φ, α ) ≡ ¬ S ( φ, α )). 3 ∀ φ ∀ ψ ∀ α ∈ Asn( φ, ψ ) ( S ( φ ∨ ψ, α ) ≡ S ( φ, α ) ∨ S ( ψ, α )). 4 ∀ φ ∀ v ∀ α ∈ Asn( ∃ v φ ) ( S ( ∃ v φ, α ) ≡ ∃ β ∼ v α S ( φ, β )). Mateusz Łełyk (IF UW) October 23, 2017, Moscow 3 / 17

Full satisfaction class Let M | = PA and S be a fresh binary predicate. A full satisfation class on M is a set S ⊆ M 2 such that the following L PA ∪ { S } sentences are true in ( M , S ): ( S ( s = t , α ) ≡ s α = t α ). 1 ∀ s , t ∀ α ∈ Asn( s , t ) 2 ∀ φ ∀ α ∈ Asn( φ ) ( S ( ¬ φ, α ) ≡ ¬ S ( φ, α )). 3 ∀ φ ∀ ψ ∀ α ∈ Asn( φ, ψ ) ( S ( φ ∨ ψ, α ) ≡ S ( φ, α ) ∨ S ( ψ, α )). 4 ∀ φ ∀ v ∀ α ∈ Asn( ∃ v φ ) ( S ( ∃ v φ, α ) ≡ ∃ β ∼ v α S ( φ, β )). 5 ∀ x , y ( S ( x , y ) → Form L PA ( x ) ∧ y ∈ Asn( x )) Mateusz Łełyk (IF UW) October 23, 2017, Moscow 3 / 17

Full satisfaction class Let M | = PA and S be a fresh binary predicate. A full satisfation class on M is a set S ⊆ M 2 such that the following L PA ∪ { S } sentences are true in ( M , S ): ( S ( s = t , α ) ≡ s α = t α ). 1 ∀ s , t ∀ α ∈ Asn( s , t ) 2 ∀ φ ∀ α ∈ Asn( φ ) ( S ( ¬ φ, α ) ≡ ¬ S ( φ, α )). 3 ∀ φ ∀ ψ ∀ α ∈ Asn( φ, ψ ) ( S ( φ ∨ ψ, α ) ≡ S ( φ, α ) ∨ S ( ψ, α )). 4 ∀ φ ∀ v ∀ α ∈ Asn( ∃ v φ ) ( S ( ∃ v φ, α ) ≡ ∃ β ∼ v α S ( φ, β )). 5 ∀ x , y ( S ( x , y ) → Form L PA ( x ) ∧ y ∈ Asn( x )) Mateusz Łełyk (IF UW) October 23, 2017, Moscow 3 / 17

Full satisfaction class Let M | = PA and S be a fresh binary predicate. A full satisfation class on M is a set S ⊆ M 2 such that the following L PA ∪ { S } sentences are true in ( M , S ): ( S ( s = t , α ) ≡ s α = t α ). 1 ∀ s , t ∀ α ∈ Asn( s , t ) 2 ∀ φ ∀ α ∈ Asn( φ ) ( S ( ¬ φ, α ) ≡ ¬ S ( φ, α )). 3 ∀ φ ∀ ψ ∀ α ∈ Asn( φ, ψ ) ( S ( φ ∨ ψ, α ) ≡ S ( φ, α ) ∨ S ( ψ, α )). 4 ∀ φ ∀ v ∀ α ∈ Asn( ∃ v φ ) ( S ( ∃ v φ, α ) ≡ ∃ β ∼ v α S ( φ, β )). 5 ∀ x , y ( S ( x , y ) → Form L PA ( x ) ∧ y ∈ Asn( x )) The theory consisting of PA and the above axioms is called CS − . Mateusz Łełyk (IF UW) October 23, 2017, Moscow 3 / 17

Full satisfaction class Let M | = PA and S be a fresh binary predicate. A full satisfation class on M is a set S ⊆ M 2 such that the following L PA ∪ { S } sentences are true in ( M , S ): ( S ( s = t , α ) ≡ s α = t α ). 1 ∀ s , t ∀ α ∈ Asn( s , t ) 2 ∀ φ ∀ α ∈ Asn( φ ) ( S ( ¬ φ, α ) ≡ ¬ S ( φ, α )). 3 ∀ φ ∀ ψ ∀ α ∈ Asn( φ, ψ ) ( S ( φ ∨ ψ, α ) ≡ S ( φ, α ) ∨ S ( ψ, α )). 4 ∀ φ ∀ v ∀ α ∈ Asn( ∃ v φ ) ( S ( ∃ v φ, α ) ≡ ∃ β ∼ v α S ( φ, β )). 5 ∀ x , y ( S ( x , y ) → Form L PA ( x ) ∧ y ∈ Asn( x )) The theory consisting of PA and the above axioms is called CS − . CS n is CS − with Σ n induction for S . Mateusz Łełyk (IF UW) October 23, 2017, Moscow 3 / 17

Full satisfaction class Let M | = PA and S be a fresh binary predicate. A full satisfation class on M is a set S ⊆ M 2 such that the following L PA ∪ { S } sentences are true in ( M , S ): ( S ( s = t , α ) ≡ s α = t α ). 1 ∀ s , t ∀ α ∈ Asn( s , t ) 2 ∀ φ ∀ α ∈ Asn( φ ) ( S ( ¬ φ, α ) ≡ ¬ S ( φ, α )). 3 ∀ φ ∀ ψ ∀ α ∈ Asn( φ, ψ ) ( S ( φ ∨ ψ, α ) ≡ S ( φ, α ) ∨ S ( ψ, α )). 4 ∀ φ ∀ v ∀ α ∈ Asn( ∃ v φ ) ( S ( ∃ v φ, α ) ≡ ∃ β ∼ v α S ( φ, β )). 5 ∀ x , y ( S ( x , y ) → Form L PA ( x ) ∧ y ∈ Asn( x )) The theory consisting of PA and the above axioms is called CS − . CS n is CS − with Σ n induction for S . CS is the sum of all CS n ’s. Mateusz Łełyk (IF UW) October 23, 2017, Moscow 3 / 17

Partial inductive satisfaction class Let M | = PA. Let S be a fresh binary predicate and c a nonstandard element of M . A partial, inductive satisfaction class ( c -restricted) on M is a set S c ⊆ M 2 such that the following L PA ∪ { S } sentences are true in ( M , S c ) ( φ ∈ compl( c ) means that φ is of complexity (logical depth) at most c ): ( S ( s = t , α ) ≡ s α = t α ). 1 ∀ s , t ∀ α ∈ Asn( s , t ) Mateusz Łełyk (IF UW) October 23, 2017, Moscow 4 / 17

Partial inductive satisfaction class Let M | = PA. Let S be a fresh binary predicate and c a nonstandard element of M . A partial, inductive satisfaction class ( c -restricted) on M is a set S c ⊆ M 2 such that the following L PA ∪ { S } sentences are true in ( M , S c ) ( φ ∈ compl( c ) means that φ is of complexity (logical depth) at most c ): ( S ( s = t , α ) ≡ s α = t α ). 1 ∀ s , t ∀ α ∈ Asn( s , t ) 2 ∀ φ ∈ compl( c ) ∀ α ∈ Asn( φ ) ( S ( ¬ φ, α ) ≡ ¬ S ( φ, α )). Mateusz Łełyk (IF UW) October 23, 2017, Moscow 4 / 17