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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 CS0 (the theory of a ∆0-inductive full satisfaction class) over ω-times iterated uniform reflection over PA (URω(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 CS0 (the theory of a ∆0-inductive full satisfaction class) over ω-times iterated uniform reflection over PA (URω(PA)). a proof of conservativity of CS0 + BΣ1(S) over URω(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 CS0 (the theory of a ∆0-inductive full satisfaction class) over ω-times iterated uniform reflection over PA (URω(PA)). a proof of conservativity of CS0 + BΣ1(S) over URω(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 ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, S):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

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 ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, S):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

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 ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, S):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

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 ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, S):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

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(∃vφ)

(S(∃vφ, α) ≡ ∃β ∼v αS(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 ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, S):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

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(∃vφ)

(S(∃vφ, α) ≡ ∃β ∼v αS(φ, β)).

5 ∀x, y

(S(x, y) → FormLPA(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 ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, S):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

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(∃vφ)

(S(∃vφ, α) ≡ ∃β ∼v αS(φ, β)).

5 ∀x, y

(S(x, y) → FormLPA(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 ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, S):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

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(∃vφ)

(S(∃vφ, α) ≡ ∃β ∼v αS(φ, β)).

5 ∀x, y

(S(x, y) → FormLPA(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 ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, S):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

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(∃vφ)

(S(∃vφ, α) ≡ ∃β ∼v αS(φ, β)).

5 ∀x, y

(S(x, y) → FormLPA(x) ∧ y ∈ Asn(x)) The theory consisting of PA and the above axioms is called CS−. CSn is CS− with Σn 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 ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, S):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

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(∃vφ)

(S(∃vφ, α) ≡ ∃β ∼v αS(φ, β)).

5 ∀x, y

(S(x, y) → FormLPA(x) ∧ y ∈ Asn(x)) The theory consisting of PA and the above axioms is called CS−. CSn is CS− with Σn induction for S. CS is the sum of all CSn’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 Sc ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, Sc) (φ ∈ compl(c) means that φ is of complexity (logical depth) at most c):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

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 Sc ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, Sc) (φ ∈ compl(c) means that φ is of complexity (logical depth) at most c):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

2 ∀φ ∈ compl(c)∀α ∈ Asn(φ)

(S(¬φ, α) ≡ ¬S(φ, α)).

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 Sc ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, Sc) (φ ∈ compl(c) means that φ is of complexity (logical depth) at most c):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

2 ∀φ ∈ compl(c)∀α ∈ Asn(φ)

(S(¬φ, α) ≡ ¬S(φ, α)).

3 ∀φ, ψ ∈ compl(c)∀α ∈ Asn(φ, ψ) (S(φ ∨ ψ, α) ≡ S(φ, α) ∨ S(ψ, α)). 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 Sc ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, Sc) (φ ∈ compl(c) means that φ is of complexity (logical depth) at most c):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

2 ∀φ ∈ compl(c)∀α ∈ Asn(φ)

(S(¬φ, α) ≡ ¬S(φ, α)).

3 ∀φ, ψ ∈ compl(c)∀α ∈ Asn(φ, ψ) (S(φ ∨ ψ, α) ≡ S(φ, α) ∨ S(ψ, α)). 4 ∀φ ∈ compl(c)∀v∀α ∈ Asn(∃vφ) (S(∃vφ, α) ≡ ∃β ∼v αS(φ, β)). 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 Sc ⊆ M2 such that the following LPA ∪ {S} sentences are true in (M, Sc) (φ ∈ compl(c) means that φ is of complexity (logical depth) at most c):

1 ∀s, t∀α ∈ Asn(s, t)

(S(s = t, α) ≡ sα = tα).

2 ∀φ ∈ compl(c)∀α ∈ Asn(φ)

(S(¬φ, α) ≡ ¬S(φ, α)).

3 ∀φ, ψ ∈ compl(c)∀α ∈ Asn(φ, ψ) (S(φ ∨ ψ, α) ≡ S(φ, α) ∨ S(ψ, α)). 4 ∀φ ∈ compl(c)∀v∀α ∈ Asn(∃vφ) (S(∃vφ, α) ≡ ∃β ∼v αS(φ, β)). 5 for every φ(x, ¯

y) in LPA ∪ {S}, the instantiation of induction scheme with φ(x, ¯ y)

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 4 / 17

Iterations of Uniform Reflection

Definition Let Th be a ∆1 definable arithmetical theory. Define UR(Th) = Th + {∀¯ x

PrTh(φ(¯

x)) → φ(¯ x)

| φ(¯

x) ∈ LPA}

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 5 / 17

Iterations of Uniform Reflection

Definition Let Th be a ∆1 definable arithmetical theory. Define UR(Th) = Th + {∀¯ x

PrTh(φ(¯

x)) → φ(¯ x)

| φ(¯

x) ∈ LPA}

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 5 / 17

Iterations of Uniform Reflection

Definition Let Th be a ∆1 definable arithmetical theory. Define UR(Th) = Th + {∀¯ x

PrTh(φ(¯

x)) → φ(¯ x)

| φ(¯

x) ∈ LPA} UR0(PA) = PA

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 5 / 17

Iterations of Uniform Reflection

Definition Let Th be a ∆1 definable arithmetical theory. Define UR(Th) = Th + {∀¯ x

PrTh(φ(¯

x)) → φ(¯ x)

| φ(¯

x) ∈ LPA} UR0(PA) = PA URn+1(PA) = UR(URn(PA))

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 5 / 17

Iterations of Uniform Reflection

Definition Let Th be a ∆1 definable arithmetical theory. Define UR(Th) = Th + {∀¯ x

PrTh(φ(¯

x)) → φ(¯ x)

| φ(¯

x) ∈ LPA} UR0(PA) = PA URn+1(PA) = UR(URn(PA)) URω(PA) =

• n∈ω

URn(PA)

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 5 / 17

Some history and context

Theorem (Enayat,Visser) If (M, Sc) is a model of PA with a partial inductive satisfaction class, then there exist M e N and a full satisfaction class S′ on N such that S ⊆ S′.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 6 / 17

Some history and context

Theorem (Enayat,Visser) If (M, Sc) is a model of PA with a partial inductive satisfaction class, then there exist M e N and a full satisfaction class S′ on N such that S ⊆ S′.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 6 / 17

Some history and context

Theorem (Enayat,Visser) If (M, Sc) is a model of PA with a partial inductive satisfaction class, then there exist M e N and a full satisfaction class S′ on N such that S ⊆ S′. However: if S is a (full or partial inductive) satisfaction class on M, then M is recursively saturated (Lachlan’s Theorem). If M is rather classless, then it does not admit a recursively saturated end-extension.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 6 / 17

Some history and context

Theorem (Enayat,Visser) If (M, Sc) is a model of PA with a partial inductive satisfaction class, then there exist M e N and a full satisfaction class S′ on N such that S ⊆ S′. However: if S is a (full or partial inductive) satisfaction class on M, then M is recursively saturated (Lachlan’s Theorem). If M is rather classless, then it does not admit a recursively saturated end-extension. Theorem (Kotlarski) CS0 is conservative over URω(PA).

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 6 / 17

Some history and context

Theorem (Enayat,Visser) If (M, Sc) is a model of PA with a partial inductive satisfaction class, then there exist M e N and a full satisfaction class S′ on N such that S ⊆ S′. However: if S is a (full or partial inductive) satisfaction class on M, then M is recursively saturated (Lachlan’s Theorem). If M is rather classless, then it does not admit a recursively saturated end-extension. Theorem (Kotlarski) CS0 is conservative over URω(PA).

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 6 / 17

Some history and context

Theorem (Enayat,Visser) If (M, Sc) is a model of PA with a partial inductive satisfaction class, then there exist M e N and a full satisfaction class S′ on N such that S ⊆ S′. However: if S is a (full or partial inductive) satisfaction class on M, then M is recursively saturated (Lachlan’s Theorem). If M is rather classless, then it does not admit a recursively saturated end-extension. Theorem (Kotlarski) CS0 is conservative over URω(PA). The proof uses countable, recursively saturated models and mimics Henkin’s proof of Completeness Theorem.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 6 / 17

Some history and context

Theorem (Enayat,Visser) If (M, Sc) is a model of PA with a partial inductive satisfaction class, then there exist M e N and a full satisfaction class S′ on N such that S ⊆ S′. However: if S is a (full or partial inductive) satisfaction class on M, then M is recursively saturated (Lachlan’s Theorem). If M is rather classless, then it does not admit a recursively saturated end-extension. Theorem (Kotlarski) CS0 is conservative over URω(PA). The proof uses countable, recursively saturated models and mimics Henkin’s proof of Completeness Theorem. It shows conservativity over ω iterations of (internal) ω-rule. That CS0 ⊢ URω(PA) was first shown by Bartosz Wcisło.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 6 / 17

Some history and context, continued

We will arithmetize the following proposition: Proposition Every model of PA has an elementary extension with a partial inductive satisfaction class.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 7 / 17

Some history and context, continued

We will arithmetize the following proposition: Proposition Every model of PA has an elementary extension with a partial inductive satisfaction class. Proof. Pick an arbitrary M | = PA, and show that ElDiag(M) is consistent with all sentences of the form

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 7 / 17

Some history and context, continued

We will arithmetize the following proposition: Proposition Every model of PA has an elementary extension with a partial inductive satisfaction class. Proof. Pick an arbitrary M | = PA, and show that ElDiag(M) is consistent with all sentences of the form

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 7 / 17

Some history and context, continued

We will arithmetize the following proposition: Proposition Every model of PA has an elementary extension with a partial inductive satisfaction class. Proof. Pick an arbitrary M | = PA, and show that ElDiag(M) is consistent with all sentences of the form ∀α (S(φ ∧ ψ, α) ≡ S(φ, α) ∧ S(ψ, α))

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 7 / 17

Some history and context, continued

We will arithmetize the following proposition: Proposition Every model of PA has an elementary extension with a partial inductive satisfaction class. Proof. Pick an arbitrary M | = PA, and show that ElDiag(M) is consistent with all sentences of the form ∀α (S(φ ∧ ψ, α) ≡ S(φ, α) ∧ S(ψ, α)) for φ ∈ LPA and full induction for S. Then use overspill over N on a formula "S is compositional for all formulae of complexity at most n."

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 7 / 17

Some context and history, continued, continued

Ending this section let me give two metamathematical properties of CS0. CS0 can be alternatively axiomatized by taking CS− with the global reflection principle, i.e. ∀φ∀α ∈ Asn(φ)

PrPA(φ) → S(φ, α) .

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 8 / 17

Some context and history, continued, continued

Ending this section let me give two metamathematical properties of CS0. CS0 can be alternatively axiomatized by taking CS− with the global reflection principle, i.e. ∀φ∀α ∈ Asn(φ)

PrPA(φ) → S(φ, α) .

CS0 has super-exponential speed-up over URω(PA). Consequently, it is non-interpretable in it.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 8 / 17

Limitations

If (N, S) | = CS0, then also (N, S) | = ∀φ∀α ∈ Asn(φ)

PrPA(φ) → S(φ, α) .

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 9 / 17

Limitations

If (N, S) | = CS0, then also (N, S) | = ∀φ∀α ∈ Asn(φ)

PrPA(φ) → S(φ, α) .

Hence if S prolongs a partial inductive satisfaction class Sc from its elementary initial segment M, then (M, Sc) | = ∀φ ∈ compl(c)∀α ∈ Asn(φ)

PrPA(φ) → S(φ, α) .

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 9 / 17

Limitations

If (N, S) | = CS0, then also (N, S) | = ∀φ∀α ∈ Asn(φ)

PrPA(φ) → S(φ, α) .

Hence if S prolongs a partial inductive satisfaction class Sc from its elementary initial segment M, then (M, Sc) | = ∀φ ∈ compl(c)∀α ∈ Asn(φ)

PrPA(φ) → S(φ, α) .

Even in models of decent theories by far not every satisfaction class has this property.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 9 / 17

Key definition

Let M | = PA, c ∈ M \ N and Sc be a partial inductive satisfaction class. We say that Sc is prolongable (1-prolongable) if there exist

1 M e N, Mateusz Łełyk (IF UW) October 23, 2017, Moscow 10 / 17

Key definition

Let M | = PA, c ∈ M \ N and Sc be a partial inductive satisfaction class. We say that Sc is prolongable (1-prolongable) if there exist

1 M e N, 2 d ∈ N \ M Mateusz Łełyk (IF UW) October 23, 2017, Moscow 10 / 17

Key definition

Let M | = PA, c ∈ M \ N and Sc be a partial inductive satisfaction class. We say that Sc is prolongable (1-prolongable) if there exist

1 M e N, 2 d ∈ N \ M 3 a partial inductive satisfaction class Sd in N which covers M. Mateusz Łełyk (IF UW) October 23, 2017, Moscow 10 / 17

Key definition

Let M | = PA, c ∈ M \ N and Sc be a partial inductive satisfaction class. We say that Sc is prolongable (1-prolongable) if there exist

1 M e N, 2 d ∈ N \ M 3 a partial inductive satisfaction class Sd in N which covers M. Mateusz Łełyk (IF UW) October 23, 2017, Moscow 10 / 17

Key definition

Let M | = PA, c ∈ M \ N and Sc be a partial inductive satisfaction class. We say that Sc is prolongable (1-prolongable) if there exist

1 M e N, 2 d ∈ N \ M 3 a partial inductive satisfaction class Sd in N which covers M.

Sc is n-prolongable if it is prolongable to an n − 1 prolongable partial inductive satisfaction class.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 10 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 11 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 11 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

2 Sc can be restricted to an n-prolongable partial inductive satisfaction

class.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 11 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

2 Sc can be restricted to an n-prolongable partial inductive satisfaction

class. Proof for the case n = 1, ⇒. If M | = UR(PA), then for every n ∈ ω we have

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 11 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

2 Sc can be restricted to an n-prolongable partial inductive satisfaction

class. Proof for the case n = 1, ⇒. If M | = UR(PA), then for every n ∈ ω we have

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 11 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

2 Sc can be restricted to an n-prolongable partial inductive satisfaction

class. Proof for the case n = 1, ⇒. If M | = UR(PA), then for every n ∈ ω we have (M, Sc) | = ∀φ ∈ compl(n)∀α ∈ Asn(φ)

PrPA(φ) → S(φ, α) .

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 11 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

2 Sc can be restricted to an n-prolongable partial inductive satisfaction

class. Proof for the case n = 1, ⇒. If M | = UR(PA), then for every n ∈ ω we have (M, Sc) | = ∀φ ∈ compl(n)∀α ∈ Asn(φ)

PrPA(φ) → S(φ, α) .

Pick a d ∈ M \ N from the overspill and work with Sd. Working in (M, Sd) take (N, S′) to be a model of the theory extending PA(S′) (i.e. we add full induction scheme for the new predicate S′) with

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 11 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

2 Sc can be restricted to an n-prolongable partial inductive satisfaction

class. Proof for the case n = 1, ⇒. If M | = UR(PA), then for every n ∈ ω we have (M, Sc) | = ∀φ ∈ compl(n)∀α ∈ Asn(φ)

PrPA(φ) → S(φ, α) .

Pick a d ∈ M \ N from the overspill and work with Sd. Working in (M, Sd) take (N, S′) to be a model of the theory extending PA(S′) (i.e. we add full induction scheme for the new predicate S′) with {S′(φ, α) | S(φ, α)} + {”S′ is compositional on φ” | φ ∈ LPA}.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 11 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 12 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 12 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

2 Sc can be restricted to an n-prolongable partial inductive satisfaction

class.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 12 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

2 Sc can be restricted to an n-prolongable partial inductive satisfaction

class. Proof for the case n = 1, ⇐. If M N, Sd prolongs Sc′ and covers M and p is a proof which involves

• nly formulae from M, then

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 12 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

2 Sc can be restricted to an n-prolongable partial inductive satisfaction

class. Proof for the case n = 1, ⇐. If M N, Sd prolongs Sc′ and covers M and p is a proof which involves

• nly formulae from M, then

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 12 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

2 Sc can be restricted to an n-prolongable partial inductive satisfaction

class. Proof for the case n = 1, ⇐. If M N, Sd prolongs Sc′ and covers M and p is a proof which involves

• nly formulae from M, then

(N, Sd) | = ∀φ ∈ p∀α ∈ Asn(φ) S(φ, α).

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 12 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

2 Sc can be restricted to an n-prolongable partial inductive satisfaction

class. Proof for the case n = 1, ⇐. If M N, Sd prolongs Sc′ and covers M and p is a proof which involves

• nly formulae from M, then

(N, Sd) | = ∀φ ∈ p∀α ∈ Asn(φ) S(φ, α). Therefore if φ(a) is standard and M | = PrPA(φ(a)), then

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 12 / 17

Proposition For arbitrary (M, Sc) and arbitrary n the following conditions are equivalent

1 M |

= URn(PA);

2 Sc can be restricted to an n-prolongable partial inductive satisfaction

class. Proof for the case n = 1, ⇐. If M N, Sd prolongs Sc′ and covers M and p is a proof which involves

• nly formulae from M, then

(N, Sd) | = ∀φ ∈ p∀α ∈ Asn(φ) S(φ, α). Therefore if φ(a) is standard and M | = PrPA(φ(a)), then (N, Sd) | = Sd(φ(a), ∅) and the same holds in (M, Sc′).

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 12 / 17

Main Theorem

Theorem For (M, Sc) the following conditions are equivalent:

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 13 / 17

Main Theorem

Theorem For (M, Sc) the following conditions are equivalent:

1 M |

= URω(PA);

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 13 / 17

Main Theorem

Theorem For (M, Sc) the following conditions are equivalent:

1 M |

= URω(PA);

2 Sc can be restricted to an infinitely prolongable satisfaction class; Mateusz Łełyk (IF UW) October 23, 2017, Moscow 13 / 17

Main Theorem

Theorem For (M, Sc) the following conditions are equivalent:

1 M |

= URω(PA);

2 Sc can be restricted to an infinitely prolongable satisfaction class; 3 there exist a restriction Sd of Sc, M e N and a full satisfaction

class S on N such that

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 13 / 17

Main Theorem

Theorem For (M, Sc) the following conditions are equivalent:

1 M |

= URω(PA);

2 Sc can be restricted to an infinitely prolongable satisfaction class; 3 there exist a restriction Sd of Sc, M e N and a full satisfaction

class S on N such that

1

(N, S) | = CS0;

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 13 / 17

Main Theorem

Theorem For (M, Sc) the following conditions are equivalent:

1 M |

= URω(PA);

2 Sc can be restricted to an infinitely prolongable satisfaction class; 3 there exist a restriction Sd of Sc, M e N and a full satisfaction

class S on N such that

1

(N, S) | = CS0;

2

Sd ⊆ S.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 13 / 17

Main Theorem

Theorem For (M, Sc) the following conditions are equivalent:

1 M |

= URω(PA);

2 Sc can be restricted to an infinitely prolongable satisfaction class; 3 there exist a restriction Sd of Sc, M e N and a full satisfaction

class S on N such that

1

(N, S) | = CS0;

2

Sd ⊆ S.

(2) ⇒ (3). Having M = M0 e M1 M2 . . . and Sd = Sd0 ⊆ Sd1 ⊆ Sd2 define N =

• i

Mi S =

• i

Sdi

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 13 / 17

Corollaries

Corollary Let M be countable and recursively saturated. Then the following are equivalent

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 14 / 17

Corollaries

Corollary Let M be countable and recursively saturated. Then the following are equivalent

1 M |

= URω(PA)

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 14 / 17

Corollaries

Corollary Let M be countable and recursively saturated. Then the following are equivalent

1 M |

= URω(PA)

2 there exist an ω + ω chain of elementary initial segments of M

M0

d0∈M0

e . . . e Mn

dn∈Mn\Mn−1

e . . . Mω

dω∈Mω\

i∈ω Mi

e Mω+1 . . . , and a full, ∆0-inductive satisfaction class S on M such that

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 14 / 17

Corollaries

Corollary Let M be countable and recursively saturated. Then the following are equivalent

1 M |

= URω(PA)

2 there exist an ω + ω chain of elementary initial segments of M

M0

d0∈M0

e . . . e Mn

dn∈Mn\Mn−1

e . . . Mω

dω∈Mω\

i∈ω Mi

e Mω+1 . . . , and a full, ∆0-inductive satisfaction class S on M such that

1

• i Mi = M.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 14 / 17

Corollaries

Corollary Let M be countable and recursively saturated. Then the following are equivalent

1 M |

= URω(PA)

2 there exist an ω + ω chain of elementary initial segments of M

M0

d0∈M0

e . . . e Mn

dn∈Mn\Mn−1

e . . . Mω

dω∈Mω\

i∈ω Mi

e Mω+1 . . . , and a full, ∆0-inductive satisfaction class S on M such that

1

• i Mi = M.

2

for each i, Sdi ∩ Mi is a partial inductive satisfaction class on Mi covering Mj for j < i.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 14 / 17

Corollaries

Corollary CS0 + BΣ1(S) is conservative over URω(PA).

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 15 / 17

Corollaries

Corollary CS0 + BΣ1(S) is conservative over URω(PA). Proof. Take M | = URω(PA) recursively saturated and countable.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 15 / 17

Corollaries

Corollary CS0 + BΣ1(S) is conservative over URω(PA). Proof. Take M | = URω(PA) recursively saturated and countable.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 15 / 17

Corollaries

Corollary CS0 + BΣ1(S) is conservative over URω(PA). Proof. Take M | = URω(PA) recursively saturated and countable. Take an ω + ω chain (Mα)α∈ω+ω, a chain of satisfaction classes (Sdα)α∈ω+ω and S as in the above corollary.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 15 / 17

Corollaries

Corollary CS0 + BΣ1(S) is conservative over URω(PA). Proof. Take M | = URω(PA) recursively saturated and countable. Take an ω + ω chain (Mα)α∈ω+ω, a chain of satisfaction classes (Sdα)α∈ω+ω and S as in the above corollary. Define Sω =

α∈ω Sdα. Then (Mω, Sω) |

= CS0 and Sω ⊆ S. Since Mω e M, (Mω, Sω) | = BΣ1(S).

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 15 / 17

Longer chains of satisfaction classes?

Thanks to Barwise-Schlipf-Ressayre theorem (countable recursively saturated models are resplendent) over an arbitrary countable recursively saturated model of URω(PA) and for arbitrary α < ω1 we can find a chain

• f satisfaction classes of length α

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 16 / 17

Longer chains of satisfaction classes?

Thanks to Barwise-Schlipf-Ressayre theorem (countable recursively saturated models are resplendent) over an arbitrary countable recursively saturated model of URω(PA) and for arbitrary α < ω1 we can find a chain

• f satisfaction classes of length α i.e. {Nβ}β<α, {Sdβ}β<α, s.t.

1 N0 = M0, |Nγ| = |Nζ| and Nγ e Nζ for γ < ζ. Mateusz Łełyk (IF UW) October 23, 2017, Moscow 16 / 17

Longer chains of satisfaction classes?

Thanks to Barwise-Schlipf-Ressayre theorem (countable recursively saturated models are resplendent) over an arbitrary countable recursively saturated model of URω(PA) and for arbitrary α < ω1 we can find a chain

• f satisfaction classes of length α i.e. {Nβ}β<α, {Sdβ}β<α, s.t.

1 N0 = M0, |Nγ| = |Nζ| and Nγ e Nζ for γ < ζ. 2 Sdγ is a partial inductive satisfaction class on Nγ covering Nζ for

ζ < γ.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 16 / 17

Longer chains of satisfaction classes?

Thanks to Barwise-Schlipf-Ressayre theorem (countable recursively saturated models are resplendent) over an arbitrary countable recursively saturated model of URω(PA) and for arbitrary α < ω1 we can find a chain

• f satisfaction classes of length α i.e. {Nβ}β<α, {Sdβ}β<α, s.t.

1 N0 = M0, |Nγ| = |Nζ| and Nγ e Nζ for γ < ζ. 2 Sdγ is a partial inductive satisfaction class on Nγ covering Nζ for

ζ < γ.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 16 / 17

Longer chains of satisfaction classes?

Thanks to Barwise-Schlipf-Ressayre theorem (countable recursively saturated models are resplendent) over an arbitrary countable recursively saturated model of URω(PA) and for arbitrary α < ω1 we can find a chain

• f satisfaction classes of length α i.e. {Nβ}β<α, {Sdβ}β<α, s.t.

1 N0 = M0, |Nγ| = |Nζ| and Nγ e Nζ for γ < ζ. 2 Sdγ is a partial inductive satisfaction class on Nγ covering Nζ for

ζ < γ. The existence of ω1 such chain is equivalent to M satisfying the arithmetical consequences of CS.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 16 / 17

Thank you for your attention.

Mateusz Łełyk (IF UW) October 23, 2017, Moscow 17 / 17