Power Kripke-Platek Set Theory, Ordinal Analysis and Global Choice

Michael Rathjen

Leverhulme Fellow

Proof Theory, Modal Logic and Reﬂection Principles Second International Wormshop

Ciudad de México, 29 September 2014

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE 1 Some iconic proof-theoretic ordinals

2 Power Kripke-Platek set theory

3 Power Kripke-Platek set theory with global choice

4 The Existence Property for intuitionistic set theories

Plan of the Talk

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE 2 Power Kripke-Platek set theory

3 Power Kripke-Platek set theory with global choice

4 The Existence Property for intuitionistic set theories

Plan of the Talk

1 Some iconic proof-theoretic ordinals

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE 3 Power Kripke-Platek set theory with global choice

4 The Existence Property for intuitionistic set theories

Plan of the Talk

1 Some iconic proof-theoretic ordinals

2 Power Kripke-Platek set theory

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE 4 The Existence Property for intuitionistic set theories

Plan of the Talk

1 Some iconic proof-theoretic ordinals

2 Power Kripke-Platek set theory

3 Power Kripke-Platek set theory with global choice

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Plan of the Talk

1 Some iconic proof-theoretic ordinals

2 Power Kripke-Platek set theory

3 Power Kripke-Platek set theory with global choice

4 The Existence Property for intuitionistic set theories

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Let T1, T2 be a pair of theories with languages L1 and L2, respectively, and let Φ be a (primitive recursive) collection of formulae common to both languages. Furthermore, Φ should contain the closed equations of the language of PRA.

T1 is proof-theoretically Φ-reducible to T2

written T1 ≤Φ T2, if there exists a primitive recursive function f such that

PRA ` ∀φ ∈ Φ ∀x [ProofT1 (x, φ) → ProofT2 (f (x), φ)]. (1)

T1 and T2 are said to be proof-theoretically Φ-equivalent, written T1 ≡Φ T2, if T1 ≤Φ T2 and T2 ≤Φ T1.

Proof-theoretic reductions

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE T1 is proof-theoretically Φ-reducible to T2

written T1 ≤Φ T2, if there exists a primitive recursive function f such that

PRA ` ∀φ ∈ Φ ∀x [ProofT1 (x, φ) → ProofT2 (f (x), φ)]. (1)

T1 and T2 are said to be proof-theoretically Φ-equivalent, written T1 ≡Φ T2, if T1 ≤Φ T2 and T2 ≤Φ T1.

Proof-theoretic reductions

Let T1, T2 be a pair of theories with languages L1 and L2, respectively, and let Φ be a (primitive recursive) collection of formulae common to both languages. Furthermore, Φ should contain the closed equations of the language of PRA.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Proof-theoretic reductions

T1 is proof-theoretically Φ-reducible to T2

written T1 ≤Φ T2, if there exists a primitive recursive function f such that

PRA ` ∀φ ∈ Φ ∀x [ProofT1 (x, φ) → ProofT2 (f (x), φ)]. (1)

T1 and T2 are said to be proof-theoretically Φ-equivalent, written T1 ≡Φ T2, if T1 ≤Φ T2 and T2 ≤Φ T1.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE • In practice, if T1 ≡ T2 is shown through an ordinal analysis this always entails that the two theories prove at least the 0 same Π2 sentences. • Given a natural ordinal representation system hA, ≺,...i of order type τ let PRA + TIqf (< τ) be PRA augmented by quantiﬁer-free induction over all initial (externally indexed) segments of ≺. • We say that a theory T has proof-theoretic ordinal τ, written |T | = τ, if T can be proof-theoretically reduced to PRA + TIqf (< τ), i.e.,

T ≡ 0 PRA + TIqf (< τ). Π2

Proof-theoretic ordinals

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE • Given a natural ordinal representation system hA, ≺,...i of order type τ let PRA + TIqf (< τ) be PRA augmented by quantiﬁer-free induction over all initial (externally indexed) segments of ≺. • We say that a theory T has proof-theoretic ordinal τ, written |T | = τ, if T can be proof-theoretically reduced to PRA + TIqf (< τ), i.e.,

T ≡ 0 PRA + TIqf (< τ). Π2

Proof-theoretic ordinals

• In practice, if T1 ≡ T2 is shown through an ordinal analysis this always entails that the two theories prove at least the 0 same Π2 sentences.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE • We say that a theory T has proof-theoretic ordinal τ, written |T | = τ, if T can be proof-theoretically reduced to PRA + TIqf (< τ), i.e.,

T ≡ 0 PRA + TIqf (< τ). Π2

Proof-theoretic ordinals

• In practice, if T1 ≡ T2 is shown through an ordinal analysis this always entails that the two theories prove at least the 0 same Π2 sentences. • Given a natural ordinal representation system hA, ≺,...i of order type τ let PRA + TIqf (< τ) be PRA augmented by quantiﬁer-free induction over all initial (externally indexed) segments of ≺.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Proof-theoretic ordinals

T ≡ 0 PRA + TIqf (< τ). Π2

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem 1 ω (i) |RCA0| = |WKL| = ω .

(ii) |ACA0| = ε0.

(iii) |ATR0| = Γ0. 1 (iv) |(Π1−CA)0|, however, eludes expression in the ordinal representations introduced so far.

Some iconic ordinals

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE ω (i) |RCA0| = |WKL| = ω .

(ii) |ACA0| = ε0.

(iii) |ATR0| = Γ0. 1 (iv) |(Π1−CA)0|, however, eludes expression in the ordinal representations introduced so far.

Some iconic ordinals

Theorem 1

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (ii) |ACA0| = ε0.

(iii) |ATR0| = Γ0. 1 (iv) |(Π1−CA)0|, however, eludes expression in the ordinal representations introduced so far.

Some iconic ordinals

Theorem 1 ω (i) |RCA0| = |WKL| = ω .

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (iii) |ATR0| = Γ0. 1 (iv) |(Π1−CA)0|, however, eludes expression in the ordinal representations introduced so far.

Some iconic ordinals

Theorem 1 ω (i) |RCA0| = |WKL| = ω .

(ii) |ACA0| = ε0.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE 1 (iv) |(Π1−CA)0|, however, eludes expression in the ordinal representations introduced so far.

Some iconic ordinals

Theorem 1 ω (i) |RCA0| = |WKL| = ω .

(ii) |ACA0| = ε0.

(iii) |ATR0| = Γ0.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Some iconic ordinals

Theorem 1 ω (i) |RCA0| = |WKL| = ω .

(ii) |ACA0| = ε0.

(iii) |ATR0| = Γ0. 1 (iv) |(Π1−CA)0|, however, eludes expression in the ordinal representations introduced so far.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem 2 The following theories have the Bachmann-Howard ordinal,

ψ (ε ) Ω1 Ω1+1 as proof-theoretic ordinal: (i) KP

(ii) ID1 (iii) BI (iv) CZF

(v) ML1V

The Bachmann-Howard ordinal

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (i) KP

(ii) ID1 (iii) BI (iv) CZF

(v) ML1V

The Bachmann-Howard ordinal

Theorem 2 The following theories have the Bachmann-Howard ordinal,

ψ (ε ) Ω1 Ω1+1 as proof-theoretic ordinal:

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (ii) ID1 (iii) BI (iv) CZF

(v) ML1V

The Bachmann-Howard ordinal

Theorem 2 The following theories have the Bachmann-Howard ordinal,

ψ (ε ) Ω1 Ω1+1 as proof-theoretic ordinal: (i) KP

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (iii) BI (iv) CZF

(v) ML1V

The Bachmann-Howard ordinal

Theorem 2 The following theories have the Bachmann-Howard ordinal,

ψ (ε ) Ω1 Ω1+1 as proof-theoretic ordinal: (i) KP

(ii) ID1

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (iv) CZF

(v) ML1V

The Bachmann-Howard ordinal

Theorem 2 The following theories have the Bachmann-Howard ordinal,

ψ (ε ) Ω1 Ω1+1 as proof-theoretic ordinal: (i) KP

(ii) ID1 (iii) BI

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (v) ML1V

The Bachmann-Howard ordinal

Theorem 2 The following theories have the Bachmann-Howard ordinal,

ψ (ε ) Ω1 Ω1+1 as proof-theoretic ordinal: (i) KP

(ii) ID1 (iii) BI (iv) CZF

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE The Bachmann-Howard ordinal

Theorem 2 The following theories have the Bachmann-Howard ordinal,

ψ (ε ) Ω1 Ω1+1 as proof-theoretic ordinal: (i) KP

(ii) ID1 (iii) BI (iv) CZF

(v) ML1V

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Kripke-Platek Set theory, KP

Though considerably weaker than ZF, a great deal of set theory requires only the axioms of KP. KP arises from ZF by completely omitting the power set axiom and restricting separation and collection to absolute predicates (cf. Barwise: admissible sets and structures (1975)), i.e. predicates deﬁnable via bounded (or ∆0) formulas. These alterations are suggested by the informal notion of ‘predicative’.

A formula is ∆0 if all its are quantiﬁers bounded, that is have one of the forms (∀x ∈b) or (∃x ∈b).

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE The Axioms of KP

Extensionality: a = b → [F(a) ↔ F(b)]. Foundation: ∀x [∀y ∈ x G(y) → G(x)] → ∀x G(x) Pair: ∃x (x = {a, b}). Union: ∃x (x = S a). Inﬁnity: ∃x x 6= ∅ ∧ (∀y ∈x)(∃z ∈x)(y ∈z). ∆0 Separation: ∃x x = {y ∈a : F(y)}

for all ∆0–formulas F.

∆0 Collection: (∀x ∈a)∃yG(x, y) → ∃z(∀x ∈a)(∃y ∈z)G(x, y)

for all ∆0–formulas G.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE P We call a formula of L∈ ∆0 if all its quantiﬁers are of the form Q x ⊆ y or Q x∈y where Q is ∀ or ∃ and x and y are distinct variables. P The ∆0 formulas are the smallest class of formulae containing the atomic formulae closed under ∧, ∨, →, ¬ and the quantiﬁers

∀x ∈ a, ∃x ∈ a, ∀x ⊆ a, ∃x ⊆ a.

KP(P) has the following axioms: Extensionality, Pairing, P Union, Inﬁnity, Powerset, ∆0 -Separation and P ∆0 -Collection.

Power Kripke-Platek Set Theory

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE P The ∆0 formulas are the smallest class of formulae containing the atomic formulae closed under ∧, ∨, →, ¬ and the quantiﬁers

∀x ∈ a, ∃x ∈ a, ∀x ⊆ a, ∃x ⊆ a.

KP(P) has the following axioms: Extensionality, Pairing, P Union, Inﬁnity, Powerset, ∆0 -Separation and P ∆0 -Collection.

Power Kripke-Platek Set Theory

P We call a formula of L∈ ∆0 if all its quantiﬁers are of the form Q x ⊆ y or Q x∈y where Q is ∀ or ∃ and x and y are distinct variables.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE KP(P) has the following axioms: Extensionality, Pairing, P Union, Inﬁnity, Powerset, ∆0 -Separation and P ∆0 -Collection.

Power Kripke-Platek Set Theory

P We call a formula of L∈ ∆0 if all its quantiﬁers are of the form Q x ⊆ y or Q x∈y where Q is ∀ or ∃ and x and y are distinct variables. P The ∆0 formulas are the smallest class of formulae containing the atomic formulae closed under ∧, ∨, →, ¬ and the quantiﬁers

∀x ∈ a, ∃x ∈ a, ∀x ⊆ a, ∃x ⊆ a.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Power Kripke-Platek Set Theory

∀x ∈ a, ∃x ∈ a, ∀x ⊆ a, ∃x ⊆ a.

KP(P) has the following axioms: Extensionality, Pairing, P Union, Inﬁnity, Powerset, ∆0 -Separation and P ∆0 -Collection.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE 1 KP(P) is not the same as KP + Powerset. The latter is a much weaker theory in which one cannot prove the existence of Vω+ω.

2 Alternatively, KP(P) can be obtained from KP by adding a function symbol P for the powerset function as a primitive symbol to the language and the axiom

∀y [y ∈ P(x) ↔ y ⊆ x]

and extending the schemes of ∆0 Separation and Collection to the ∆0 formulae of this new language.

3 The power admissible sets are the transitive models of KP(P).

Remark.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE 2 Alternatively, KP(P) can be obtained from KP by adding a function symbol P for the powerset function as a primitive symbol to the language and the axiom

∀y [y ∈ P(x) ↔ y ⊆ x]

and extending the schemes of ∆0 Separation and Collection to the ∆0 formulae of this new language.

3 The power admissible sets are the transitive models of KP(P).

Remark.

1 KP(P) is not the same as KP + Powerset. The latter is a much weaker theory in which one cannot prove the existence of Vω+ω.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE 3 The power admissible sets are the transitive models of KP(P).

Remark.

1 KP(P) is not the same as KP + Powerset. The latter is a much weaker theory in which one cannot prove the existence of Vω+ω.

2 Alternatively, KP(P) can be obtained from KP by adding a function symbol P for the powerset function as a primitive symbol to the language and the axiom

∀y [y ∈ P(x) ↔ y ⊆ x]

and extending the schemes of ∆0 Separation and Collection to the ∆0 formulae of this new language.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Remark.

1 KP(P) is not the same as KP + Powerset. The latter is a much weaker theory in which one cannot prove the existence of Vω+ω.

2 Alternatively, KP(P) can be obtained from KP by adding a function symbol P for the powerset function as a primitive symbol to the language and the axiom

∀y [y ∈ P(x) ↔ y ⊆ x]

and extending the schemes of ∆0 Separation and Collection to the ∆0 formulae of this new language.

3 The power admissible sets are the transitive models of KP(P).

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Here is an example of a structure which is a model of

KP + Powerset

but not of KP(P):

L (ℵω)L

Example

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE L (ℵω)L

Example

Here is an example of a structure which is a model of

KP + Powerset

but not of KP(P):

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Example

Here is an example of a structure which is a model of

KP + Powerset

but not of KP(P):

L (ℵω)L

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem: (H.Friedman 1973) KP(P) + AC does not prove the existence of a non-recursive von Neumann ordinal.

Proof uses Barwise compactness and truncation.

Theorem: (Mathias 2001) KP(P) + V = L proves the consistency of KP(P).

Proof uses forcing in the context of non-standard models of KP(P) and other techniques.

Older work

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem: (Mathias 2001) KP(P) + V = L proves the consistency of KP(P).

Proof uses forcing in the context of non-standard models of KP(P) and other techniques.

Older work

Theorem: (H.Friedman 1973) KP(P) + AC does not prove the existence of a non-recursive von Neumann ordinal.

Proof uses Barwise compactness and truncation.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Older work

Theorem: (H.Friedman 1973) KP(P) + AC does not prove the existence of a non-recursive von Neumann ordinal.

Proof uses Barwise compactness and truncation.

Theorem: (Mathias 2001) KP(P) + V = L proves the consistency of KP(P).

Proof uses forcing in the context of non-standard models of KP(P) and other techniques.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE KP(P) is not quite the same as the theory

KPP

in Mathias’ 2001 paper. The difference between KP(P) and KPP is that in the latter P system set induction only holds for Σ1 -formulae, or what P amounts to the same, Π1 foundation A 6= ∅ → ∃x ∈ A x ∩ A = ∅

P for Π1 classes A.

Warning

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE The difference between KP(P) and KPP is that in the latter P system set induction only holds for Σ1 -formulae, or what P amounts to the same, Π1 foundation A 6= ∅ → ∃x ∈ A x ∩ A = ∅

P for Π1 classes A.

Warning

KP(P) is not quite the same as the theory

KPP

in Mathias’ 2001 paper.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Warning

KP(P) is not quite the same as the theory

KPP

P for Π1 classes A.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE The techniques used for the ordinal analysis of KP can be adapted to yield the following result about KP(P) + AC:

Theorem: P IfA is a Π2 -formula and KP(P) + AC ` A then

VψΩ(εΩ+1) |= A.

The bound of this Theorem is sharp, that is, ψΩ(εΩ+1) is the ﬁrst ordinal with that property.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE P P We deﬁne the RSΩ –terms. To each RSΩ –term t we also assign its level, |t|. P 1. For each α < Ω, Vα is an RSΩ –term with | Vα | = α. 2. For each α < Ω, we have inﬁnitely many free variables α α α P α a1 , a2 , a3 ,... which are RSΩ –terms with | ai | = α. ~ P 3. If F(x, y ) is a ∆0 formula (whose free variables are exactly those indicated) and ~s ≡ s1, ··· , sn are P RSΩ –terms, then the formal expression

{x∈Vα | F(x,~s )}

P is an RSΩ –term with | {x∈Vα | F(x,~s )} | = α.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (i) KP(P)

(ii) CZF + Powerset (Basically IZF with Bounded Separation)

Same strength

Theorem: (R. 2012) The following theories have the same proof-theoretic strength

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (ii) CZF + Powerset (Basically IZF with Bounded Separation)

Same strength

Theorem: (R. 2012) The following theories have the same proof-theoretic strength (i) KP(P)

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Same strength

Theorem: (R. 2012) The following theories have the same proof-theoretic strength (i) KP(P)

(ii) CZF + Powerset (Basically IZF with Bounded Separation)

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem: P P KP + V = L proves the consistency, actually the Σ1 soundness, of KP(P) + AC.

This follows from the ordinal analysis of KP(P) + AC and the fact that KPP + V = L proves the existence of the Bachmann-Howard ordinal (as a set-theoretic ordinal). This strengthens Mathias’ result and also provides an entirely different proof.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE This follows from the ordinal analysis of KP(P) + AC and the fact that KPP + V = L proves the existence of the Bachmann-Howard ordinal (as a set-theoretic ordinal). This strengthens Mathias’ result and also provides an entirely different proof.

Theorem: P P KP + V = L proves the consistency, actually the Σ1 soundness, of KP(P) + AC.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE This strengthens Mathias’ result and also provides an entirely different proof.

Theorem: P P KP + V = L proves the consistency, actually the Σ1 soundness, of KP(P) + AC.

This follows from the ordinal analysis of KP(P) + AC and the fact that KPP + V = L proves the existence of the Bachmann-Howard ordinal (as a set-theoretic ordinal).

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem: P P KP + V = L proves the consistency, actually the Σ1 soundness, of KP(P) + AC.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem: KP + Powerset + V = L and KP + Powerset have the same strength.

What about the strength of KP + Powerset + V = L?

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE What about the strength of KP + Powerset + V = L?

Theorem: KP + Powerset + V = L and KP + Powerset have the same strength.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (a) ∀x [x 6= ∅ → ∃y ∈ x R(x, y)] (b) ∀x∀y∀z [R(x, y) ∧ R(x, z) → y = z] (c) Extend schemata of KP(P) to new language.

Does KP(P) + AC have the same proof-theoretic strength as KP(P)?

Let GAC be the axiom of global choice. For instance, add new two place predicate symbol R to the language and the following axioms:

Mathias’ question

Let GAC be the axiom of global choice. For instance, add new two place predicate symbol R to the language and the following axioms:

Mathias’ question

Does KP(P) + AC have the same proof-theoretic strength as KP(P)?

For instance, add new two place predicate symbol R to the language and the following axioms:

Mathias’ question

Does KP(P) + AC have the same proof-theoretic strength as KP(P)?

Let GAC be the axiom of global choice.

Mathias’ question

Does KP(P) + AC have the same proof-theoretic strength as KP(P)?

Let GAC be the axiom of global choice. For instance, add new two place predicate symbol R to the language and the following axioms:

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (b) ∀x∀y∀z [R(x, y) ∧ R(x, z) → y = z] (c) Extend schemata of KP(P) to new language.

Mathias’ question

Does KP(P) + AC have the same proof-theoretic strength as KP(P)?

Let GAC be the axiom of global choice. For instance, add new two place predicate symbol R to the language and the following axioms: (a) ∀x [x 6= ∅ → ∃y ∈ x R(x, y)]

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (c) Extend schemata of KP(P) to new language.

Mathias’ question

Does KP(P) + AC have the same proof-theoretic strength as KP(P)?

Let GAC be the axiom of global choice. For instance, add new two place predicate symbol R to the language and the following axioms: (a) ∀x [x 6= ∅ → ∃y ∈ x R(x, y)] (b) ∀x∀y∀z [R(x, y) ∧ R(x, z) → y = z]

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Mathias’ question

Does KP(P) + AC have the same proof-theoretic strength as KP(P)?

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem: If KP(P) + GAC ` θ where θ is a ΣP -sentence, then one can explicitly ﬁnd an ordinal (notation) τ < ψΩ(εΩ+1) such that

KP+AC+the von Neumann hierarchy (Vα)α≤τ exists proves θ.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem: Let τ be a limit ordinal. If

KP+AC+the von Neumann hierarchy (Vα)α<τ exists

1 proves a Π4 statements Φ of second order arithmetic, then

Z + the von Neumann hierarchy (Vα)α<τ·4+4 exists proves Φ.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Corollary: 1 If Φ is Π4 sentence such that KP(P) + GAC ` Φ then KP(P) ` Φ.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (i) KP(P) (ii) KP(P) + GAC. (iii) CZF + Powerset (iv) CZF + AC

(v) ML1Prop. (vi) CZF + Pow¬¬ (vii) OST(P)

(viii) Z + {‘Vτ exists’}τ∈BH

Same strength

Theorem: The following have the same proof-theoretic strength

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (ii) KP(P) + GAC. (iii) CZF + Powerset (iv) CZF + AC

(v) ML1Prop. (vi) CZF + Pow¬¬ (vii) OST(P)

(viii) Z + {‘Vτ exists’}τ∈BH

Same strength

Theorem: The following have the same proof-theoretic strength (i) KP(P)

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (iii) CZF + Powerset (iv) CZF + AC

(v) ML1Prop. (vi) CZF + Pow¬¬ (vii) OST(P)

(viii) Z + {‘Vτ exists’}τ∈BH

Same strength

Theorem: The following have the same proof-theoretic strength (i) KP(P) (ii) KP(P) + GAC.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (iv) CZF + AC

(v) ML1Prop. (vi) CZF + Pow¬¬ (vii) OST(P)

(viii) Z + {‘Vτ exists’}τ∈BH

Same strength

Theorem: The following have the same proof-theoretic strength (i) KP(P) (ii) KP(P) + GAC. (iii) CZF + Powerset

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (v) ML1Prop. (vi) CZF + Pow¬¬ (vii) OST(P)

(viii) Z + {‘Vτ exists’}τ∈BH

Same strength

Theorem: The following have the same proof-theoretic strength (i) KP(P) (ii) KP(P) + GAC. (iii) CZF + Powerset (iv) CZF + AC

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (vi) CZF + Pow¬¬ (vii) OST(P)

(viii) Z + {‘Vτ exists’}τ∈BH

Same strength

Theorem: The following have the same proof-theoretic strength (i) KP(P) (ii) KP(P) + GAC. (iii) CZF + Powerset (iv) CZF + AC

(v) ML1Prop.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (vii) OST(P)

(viii) Z + {‘Vτ exists’}τ∈BH

Same strength

Theorem: The following have the same proof-theoretic strength (i) KP(P) (ii) KP(P) + GAC. (iii) CZF + Powerset (iv) CZF + AC

(v) ML1Prop. (vi) CZF + Pow¬¬

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (vii) OST(P)

(viii) Z + {‘Vτ exists’}τ∈BH

Same strength

Theorem: The following have the same proof-theoretic strength (i) KP(P) (ii) KP(P) + GAC. (iii) CZF + Powerset (iv) CZF + AC

(v) ML1Prop. (vi) CZF + Pow¬¬

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE (viii) Z + {‘Vτ exists’}τ∈BH

Same strength

Theorem: The following have the same proof-theoretic strength (i) KP(P) (ii) KP(P) + GAC. (iii) CZF + Powerset (iv) CZF + AC

(v) ML1Prop. (vi) CZF + Pow¬¬ (vii) OST(P)

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Same strength

Theorem: The following have the same proof-theoretic strength (i) KP(P) (ii) KP(P) + GAC. (iii) CZF + Powerset (iv) CZF + AC

(v) ML1Prop. (vi) CZF + Pow¬¬ (vii) OST(P)

(viii) Z + {‘Vτ exists’}τ∈BH

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE 1 T has the numerical existence property, NEP, if whenever T ` (∃x∈ω)φ(x) holds for a formula φ(x) with at most the free variable x, then T ` φ(n¯) for some n. 2 T has the existence property, EP, if whenever T ` ∃xφ(x) holds for a formula φ(x) having at most the free variable x, then there is a formula ϑ(x) with exactly x free, so that T ` ∃!x ϑ(x) and T ` ∃x [ϑ(x) ∧ φ(x)].

The Existence Property

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE 2 T has the existence property, EP, if whenever T ` ∃xφ(x) holds for a formula φ(x) having at most the free variable x, then there is a formula ϑ(x) with exactly x free, so that T ` ∃!x ϑ(x) and T ` ∃x [ϑ(x) ∧ φ(x)].

The Existence Property

1 T has the numerical existence property, NEP, if whenever T ` (∃x∈ω)φ(x) holds for a formula φ(x) with at most the free variable x, then T ` φ(n¯) for some n.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE The Existence Property

1 T has the numerical existence property, NEP, if whenever T ` (∃x∈ω)φ(x) holds for a formula φ(x) with at most the free variable x, then T ` φ(n¯) for some n. 2 T has the existence property, EP, if whenever T ` ∃xφ(x) holds for a formula φ(x) having at most the free variable x, then there is a formula ϑ(x) with exactly x free, so that T ` ∃!x ϑ(x) and T ` ∃x [ϑ(x) ∧ φ(x)].

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE * Extensionality • Pairing, Union, Inﬁnity • Full Separation • Powerset # Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b (∀x ∈ a)(∃y ∈ b) ϕ(x, y)

* Set Induction

(IND∈) ∀a (∀x ∈ a ϕ(x) → ϕ(a)) → ∀a ϕ(a),

Intuitionistic Zermelo-Fraenkel set theory, IZF

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE • Pairing, Union, Inﬁnity • Full Separation • Powerset # Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b (∀x ∈ a)(∃y ∈ b) ϕ(x, y)

* Set Induction

(IND∈) ∀a (∀x ∈ a ϕ(x) → ϕ(a)) → ∀a ϕ(a),

Intuitionistic Zermelo-Fraenkel set theory, IZF

* Extensionality

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE • Full Separation • Powerset # Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b (∀x ∈ a)(∃y ∈ b) ϕ(x, y)

* Set Induction

(IND∈) ∀a (∀x ∈ a ϕ(x) → ϕ(a)) → ∀a ϕ(a),

Intuitionistic Zermelo-Fraenkel set theory, IZF

* Extensionality • Pairing, Union, Inﬁnity

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE # Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b (∀x ∈ a)(∃y ∈ b) ϕ(x, y)

* Set Induction

(IND∈) ∀a (∀x ∈ a ϕ(x) → ϕ(a)) → ∀a ϕ(a),

Intuitionistic Zermelo-Fraenkel set theory, IZF

* Extensionality • Pairing, Union, Inﬁnity • Full Separation • Powerset

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE * Set Induction

(IND∈) ∀a (∀x ∈ a ϕ(x) → ϕ(a)) → ∀a ϕ(a),

Intuitionistic Zermelo-Fraenkel set theory, IZF

* Extensionality • Pairing, Union, Inﬁnity • Full Separation • Powerset # Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b (∀x ∈ a)(∃y ∈ b) ϕ(x, y)

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Intuitionistic Zermelo-Fraenkel set theory, IZF

* Extensionality • Pairing, Union, Inﬁnity • Full Separation • Powerset # Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b (∀x ∈ a)(∃y ∈ b) ϕ(x, y)

* Set Induction

(IND∈) ∀a (∀x ∈ a ϕ(x) → ϕ(a)) → ∀a ϕ(a),

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE * Extensionality • Pairing, Union, Inﬁnity • Bounded Separation • Exponentiation # Strong Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b [(∀x ∈ a)(∃y ∈ b) ϕ(x, y) ∧ (∀y ∈ b)(∃x ∈ a) ϕ(x, y)]

* Set Induction scheme

CZF− is CZF without Exponentiation.

Constructive Zermelo-Fraenkel set theory, CZF

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE • Pairing, Union, Inﬁnity • Bounded Separation • Exponentiation # Strong Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b [(∀x ∈ a)(∃y ∈ b) ϕ(x, y) ∧ (∀y ∈ b)(∃x ∈ a) ϕ(x, y)]

* Set Induction scheme

CZF− is CZF without Exponentiation.

Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE • Bounded Separation • Exponentiation # Strong Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b [(∀x ∈ a)(∃y ∈ b) ϕ(x, y) ∧ (∀y ∈ b)(∃x ∈ a) ϕ(x, y)]

* Set Induction scheme

CZF− is CZF without Exponentiation.

Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality • Pairing, Union, Inﬁnity

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE • Exponentiation # Strong Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b [(∀x ∈ a)(∃y ∈ b) ϕ(x, y) ∧ (∀y ∈ b)(∃x ∈ a) ϕ(x, y)]

* Set Induction scheme

CZF− is CZF without Exponentiation.

Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality • Pairing, Union, Inﬁnity • Bounded Separation

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE # Strong Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b [(∀x ∈ a)(∃y ∈ b) ϕ(x, y) ∧ (∀y ∈ b)(∃x ∈ a) ϕ(x, y)]

* Set Induction scheme

CZF− is CZF without Exponentiation.

Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality • Pairing, Union, Inﬁnity • Bounded Separation • Exponentiation

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE * Set Induction scheme

CZF− is CZF without Exponentiation.

Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality • Pairing, Union, Inﬁnity • Bounded Separation • Exponentiation # Strong Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b [(∀x ∈ a)(∃y ∈ b) ϕ(x, y) ∧ (∀y ∈ b)(∃x ∈ a) ϕ(x, y)]

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE CZF− is CZF without Exponentiation.

Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality • Pairing, Union, Inﬁnity • Bounded Separation • Exponentiation # Strong Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b [(∀x ∈ a)(∃y ∈ b) ϕ(x, y) ∧ (∀y ∈ b)(∃x ∈ a) ϕ(x, y)]

* Set Induction scheme

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality • Pairing, Union, Inﬁnity • Bounded Separation • Exponentiation # Strong Collection

(∀x ∈ a) ∃y ϕ(x, y) → ∃b [(∀x ∈ a)(∃y ∈ b) ϕ(x, y) ∧ (∀y ∈ b)(∃x ∈ a) ϕ(x, y)]

* Set Induction scheme

CZF− is CZF without Exponentiation.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem.( Friedman, Šcedrovˇ 1985) IZF does not have the existence property. • (Beeson 1985) Does any reasonable set theory with Collection have the existential deﬁnability property?

Problems

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE • (Beeson 1985) Does any reasonable set theory with Collection have the existential deﬁnability property?

Problems

Theorem.( Friedman, Šcedrovˇ 1985) IZF does not have the existence property.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Problems

Theorem.( Friedman, Šcedrovˇ 1985) IZF does not have the existence property. • (Beeson 1985) Does any reasonable set theory with Collection have the existential deﬁnability property?

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE T has the weak existence property, wEP, if whenever

T ` ∃xφ(x)

holds for a formula φ(x) having at most the free variable x, then there is a formula ϑ(x) with exactly x free, so that

T ` ∃!x ϑ(x), T ` ∀x [ϑ(x) → ∃u u ∈ x], T ` ∀x [ϑ(x) → ∀u ∈ x φ(u)].

The Weak Existence Property

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE The Weak Existence Property

T has the weak existence property, wEP, if whenever

T ` ∃xφ(x)

holds for a formula φ(x) having at most the free variable x, then there is a formula ϑ(x) with exactly x free, so that

T ` ∃!x ϑ(x), T ` ∀x [ϑ(x) → ∃u u ∈ x], T ` ∀x [ϑ(x) → ∀u ∈ x φ(u)].

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem IZF does not have the weak existence property property.

IZF and wEP

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE IZF and wEP

Theorem IZF does not have the weak existence property property.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE T has the uniform weak existence property, uwEP, if the following holds: if

T ` ∀u ∃xA(u, x)

holds for a formula A(u, x) having at most the free variables u, x, then there is a formula B(u, x) with exactly u, x free, so that

T ` ∀u ∃!x B(u, x), T ` ∀u ∀x [B(u, x) → ∃z z ∈ x], T ` ∀u ∀x [B(u, x) → ∀z ∈ x A(u, z)].

The Uniform Weak Existence Property

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE The Uniform Weak Existence Property

T has the uniform weak existence property, uwEP, if the following holds: if

T ` ∀u ∃xA(u, x)

holds for a formula A(u, x) having at most the free variables u, x, then there is a formula B(u, x) with exactly u, x free, so that

T ` ∀u ∃!x B(u, x), T ` ∀u ∀x [B(u, x) → ∃z z ∈ x], T ` ∀u ∀x [B(u, x) → ∀z ∈ x A(u, z)].

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem The theories CZF−, CZF and CZF + Pow have the uniform weak existence property.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE • THEOREM If CZF ` ∃x A(x) then one can effectively construct a ΣE formula B(y) such that

CZF ` ∃!y B(y)

CZF ` ∀y[ B(y) → ∃x x ∈ y]

CZF ` ∀y [B(y) → ∀x ∈ y A(x)]

Even better

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Even better

• THEOREM If CZF ` ∃x A(x) then one can effectively construct a ΣE formula B(y) such that

CZF ` ∃!y B(y)

CZF ` ∀y[ B(y) → ∃x x ∈ y]

CZF ` ∀y [B(y) → ∀x ∈ y A(x)]

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE • THEOREM If CZF + Pow ` ∃x A(x) then one can effectively construct a ΣP formula B(y) such that

CZF + Pow ` ∃!y B(y)

CZF + Pow ` ∀y[ B(y) → ∃x x ∈ y]

CZF + Pow ` ∀y [B(y) → ∀x ∈ y A(x)]

Even better

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Even better

• THEOREM If CZF + Pow ` ∃x A(x) then one can effectively construct a ΣP formula B(y) such that

CZF + Pow ` ∃!y B(y)

CZF + Pow ` ∀y[ B(y) → ∃x x ∈ y]

CZF + Pow ` ∀y [B(y) → ∀x ∈ y A(x)]

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE IKP(P) is intuitionistic Power Kripke-Platek Set Theory. E We call a formula of L∈ ∆0 if all its quantiﬁers are of the form Q x ∈ ba or Q x∈a where Q is ∀ or ∃. IKP(E) is the intuitionistic theory with the axioms: Extensionality, Pairing, Union, Inﬁnity, Exponentiation, E E ∆0 -Separation and ∆0 -Collection.

Intuitionistic Power Kripke-Platek Set Theory and friends

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE E We call a formula of L∈ ∆0 if all its quantiﬁers are of the form Q x ∈ ba or Q x∈a where Q is ∀ or ∃. IKP(E) is the intuitionistic theory with the axioms: Extensionality, Pairing, Union, Inﬁnity, Exponentiation, E E ∆0 -Separation and ∆0 -Collection.

Intuitionistic Power Kripke-Platek Set Theory and friends

IKP(P) is intuitionistic Power Kripke-Platek Set Theory.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE IKP(E) is the intuitionistic theory with the axioms: Extensionality, Pairing, Union, Inﬁnity, Exponentiation, E E ∆0 -Separation and ∆0 -Collection.

Intuitionistic Power Kripke-Platek Set Theory and friends

IKP(P) is intuitionistic Power Kripke-Platek Set Theory. E We call a formula of L∈ ∆0 if all its quantiﬁers are of the form Q x ∈ ba or Q x∈a where Q is ∀ or ∃.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Intuitionistic Power Kripke-Platek Set Theory and friends

IKP(P) is intuitionistic Power Kripke-Platek Set Theory. E We call a formula of L∈ ∆0 if all its quantiﬁers are of the form Q x ∈ ba or Q x∈a where Q is ∀ or ∃. IKP(E) is the intuitionistic theory with the axioms: Extensionality, Pairing, Union, Inﬁnity, Exponentiation, E E ∆0 -Separation and ∆0 -Collection.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE THEOREM CZF is conservative over IKP(E) for ΣE sentences.

THEOREM CZF + Pow is conservative over IKP(P) for ΣP sentences.

Conservativity

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE THEOREM CZF + Pow is conservative over IKP(P) for ΣP sentences.

Conservativity

THEOREM CZF is conservative over IKP(E) for ΣE sentences.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Conservativity

THEOREM CZF is conservative over IKP(E) for ΣE sentences.

THEOREM CZF + Pow is conservative over IKP(P) for ΣP sentences.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem 1: IKP has the existence property for Σ1 formulae.

E Theorem 2: IKP(E) has the existence property for Σ1 formulae

P Theorem 3: IKP(P) has the existence property for Σ1 formulae.

Existence property I

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE E Theorem 2: IKP(E) has the existence property for Σ1 formulae

P Theorem 3: IKP(P) has the existence property for Σ1 formulae.

Existence property I

Theorem 1: IKP has the existence property for Σ1 formulae.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE P Theorem 3: IKP(P) has the existence property for Σ1 formulae.

Existence property I

Theorem 1: IKP has the existence property for Σ1 formulae.

E Theorem 2: IKP(E) has the existence property for Σ1 formulae

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Existence property I

Theorem 1: IKP has the existence property for Σ1 formulae.

E Theorem 2: IKP(E) has the existence property for Σ1 formulae

P Theorem 3: IKP(P) has the existence property for Σ1 formulae.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem 1: CZF− has the existence property.

Theorem 2: CZF has the existence property.

Theorem 3: CZF + Pow has the existence property.

Theorem 4:( A. Swan) CZF + Subset Collection does not have the weak existence property.

Existence property II

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem 2: CZF has the existence property.

Theorem 3: CZF + Pow has the existence property.

Theorem 4:( A. Swan) CZF + Subset Collection does not have the weak existence property.

Existence property II

Theorem 1: CZF− has the existence property.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem 3: CZF + Pow has the existence property.

Theorem 4:( A. Swan) CZF + Subset Collection does not have the weak existence property.

Existence property II

Theorem 1: CZF− has the existence property.

Theorem 2: CZF has the existence property.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Theorem 4:( A. Swan) CZF + Subset Collection does not have the weak existence property.

Existence property II

Theorem 1: CZF− has the existence property.

Theorem 2: CZF has the existence property.

Theorem 3: CZF + Pow has the existence property.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Existence property II

Theorem 1: CZF− has the existence property.

Theorem 2: CZF has the existence property.

Theorem 3: CZF + Pow has the existence property.

Theorem 4:( A. Swan) CZF + Subset Collection does not have the weak existence property.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Muchas Gracias

Finis operis

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE Finis operis

Muchas Gracias

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE References

1 M . Rathjen: From the weak to the strong existence property, Annals of Pure and Applied Logic 163 (2012) 1400-1418.

2 M. Rathjen: Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions. In: Epistemology versus Ontology: Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf, (Springer, Dordrecht, Heidelberg, 2012) 313–349.

3 M. Rathjen: Relativized ordinal analysis: The case of Power Kripke-Platek set theory. Annals of Pure and Applied Logic 165 (2014) 316339.

4 A.W. Swan: CZF does not have the existence property. Annals of Pure and Applied Logic 165 (2014) 1115–1147.

POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS,GLOBAL CHOICE