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Power Kripke-Platek Set Theory, Ordinal Analysisand Global Choice

Michael Rathjen

Leverhulme Fellow

Proof Theory, Modal Logic and ReflectionPrinciples

Second International Wormshop

Ciudad de México, 29 September 2014

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

Plan of the Talk

Proof-theoretic reductions

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

T1 is proof-theoretically Φ-reducible to T2

written T1 ≤Φ T2, if there exists a primitive recursivefunction 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.

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

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

Proof-theoretic ordinals

• In practice, if T1 ≡ T2 is shown through an ordinal analysisthis always entails that the two theories prove at least thesame Π0

2 sentences.

• Given a natural ordinal representation system 〈A,≺, . . .〉 oforder type τ let PRA + TIqf (< τ) be PRA augmented byquantifier-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 toPRA + TIqf (< τ), i.e.,

T ≡Π02

PRA + TIqf (< τ).

2 sentences.

T ≡Π02

PRA + TIqf (< τ).

2 sentences.

T ≡Π02

PRA + TIqf (< τ).

2 sentences.

T ≡Π02

PRA + TIqf (< τ).

Some iconic ordinals

Theorem 1

(i) |RCA0| = |WKL| = ωω.

(ii) |ACA0| = ε0.

(iii) |ATR0| = Γ0.

(iv) |(Π11−CA)0|, however, eludes expression in the ordinal

representations introduced so far.

Theorem 1

(i) |RCA0| = |WKL| = ωω.

(ii) |ACA0| = ε0.

(iii) |ATR0| = Γ0.

Theorem 1

(i) |RCA0| = |WKL| = ωω.

(ii) |ACA0| = ε0.

(iii) |ATR0| = Γ0.

Theorem 1

(i) |RCA0| = |WKL| = ωω.

(ii) |ACA0| = ε0.

(iii) |ATR0| = Γ0.

Theorem 1

(i) |RCA0| = |WKL| = ωω.

(ii) |ACA0| = ε0.

(iii) |ATR0| = Γ0.

Theorem 1

(i) |RCA0| = |WKL| = ωω.

(ii) |ACA0| = ε0.

(iii) |ATR0| = Γ0.

The Bachmann-Howard ordinal

Theorem 2 The following theories have theBachmann-Howard ordinal,

(εΩ1+1)

as proof-theoretic ordinal:

(i) KP

(ii) ID1

(iii) BI

(iv) CZF

(v) ML1V

(εΩ1+1)

(i) KP

(ii) ID1

(iii) BI

(iv) CZF

(v) ML1V

(εΩ1+1)

(i) KP

(ii) ID1

(iii) BI

(iv) CZF

(v) ML1V

(εΩ1+1)

(i) KP

(ii) ID1

(iii) BI

(iv) CZF

(v) ML1V

(εΩ1+1)

(i) KP

(ii) ID1

(iii) BI

(iv) CZF

(v) ML1V

(εΩ1+1)

(i) KP

(ii) ID1

(iii) BI

(iv) CZF

(v) ML1V

(εΩ1+1)

(i) KP

(ii) ID1

(iii) BI

(iv) CZF

(v) ML1V

Kripke-Platek Set theory, KP

Though considerably weaker than ZF, a great deal of set theoryrequires only the axioms of KP. KP arises from ZF bycompletely omitting the power set axiom and restrictingseparation and collection to absolute predicates (cf. Barwise:admissible sets and structures (1975)), i.e. predicates definablevia bounded (or ∆0) formulas. These alterations are suggestedby the informal notion of ‘predicative’.

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

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 =⋃

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 Kripke-Platek Set Theory

We call a formula of L∈ ∆P0 if all its quantifiers are of theform Q x ⊆ y or Q x∈y where Q is ∀ or ∃ and x and y aredistinct variables.The ∆P0 formulas are the smallest class of formulaecontaining the atomic formulae closed under ∧,∨,→,¬and the quantifiers

∀x ∈ a, ∃x ∈ a, ∀x ⊆ a, ∃x ⊆ a.

KP(P) has the following axioms: Extensionality, Pairing,Union, Infinity, Powerset, ∆P0 -Separation and∆P0 -Collection.

We call a formula of L∈ ∆P0 if all its quantifiers are of theform Q x ⊆ y or Q x∈y where Q is ∀ or ∃ and x and y aredistinct variables.

The ∆P0 formulas are the smallest class of formulaecontaining the atomic formulae closed under ∧,∨,→,¬and the quantifiers

∀x ∈ a, ∃x ∈ a, ∀x ⊆ a, ∃x ⊆ a.

Remark.

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

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

∀y [y ∈ P(x)↔ y ⊆ x ]

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

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

Remark.

∀y [y ∈ P(x)↔ y ⊆ x ]

Remark.

∀y [y ∈ P(x)↔ y ⊆ x ]

Remark.

∀y [y ∈ P(x)↔ y ⊆ x ]

Example

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

KP + Powerset

but not of KP(P):

L(ℵω)L

Example

KP + Powerset

but not of KP(P):

L(ℵω)L

Example

KP + Powerset

but not of KP(P):

L(ℵω)L

Older work

Theorem:(H.Friedman 1973)

KP(P) + AC does not prove the existence of a non-recursivevon 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 ofKP(P) and other techniques.

Older work

Warning

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

in Mathias’ 2001 paper.

The difference between KP(P) and KPP is that in the lattersystem set induction only holds for ΣP1 -formulae, or whatamounts to the same, ΠP1 foundation

A 6= ∅ → ∃x ∈ A x ∩ A = ∅

for ΠP1 classes A.

Warning

A 6= ∅ → ∃x ∈ A x ∩ A = ∅

for ΠP1 classes A.

Warning

A 6= ∅ → ∃x ∈ A x ∩ A = ∅

for ΠP1 classes A.

The techniques used for the ordinal analysis of KP can beadapted to yield the following result about KP(P) + AC:

Theorem:

If A is a ΠP2 -formula and

KP(P) + AC ` A

thenVψΩ(εΩ+1) |= A.

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

We define the RSPΩ –terms. To each RSPΩ –term t we alsoassign its level, |t |.

1. For each α < Ω, Vα is an RSPΩ –term with |Vα | = α.

2. For each α < Ω, we have infinitely many free variablesaα1 ,a

α2 ,a

α3 , . . . which are RSPΩ –terms with |aαi | = α.

3. If F (x , ~y ) is a ∆P0 formula (whose free variables areexactly those indicated) and ~s ≡ s1, · · · , sn areRSPΩ –terms, then the formal expression

x∈Vα | F (x ,~s )

is an RSPΩ –term with | x∈Vα | F (x ,~s ) | = α.

Same strength

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

(i) KP(P)

(ii) CZF + Powerset

(Basically IZF with Bounded Separation)

Same strength

(i) KP(P)

(ii) CZF + Powerset

Same strength

(i) KP(P)

(ii) CZF + Powerset

Theorem:

KPP + V = L proves the consistency, actually the ΣP1soundness, of KP(P) + AC.

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

This strengthens Mathias’ result and also provides anentirely different proof.

Theorem:

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

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

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

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

Mathias’ question

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

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

(a) ∀x [x 6= ∅ → ∃y ∈ x R(x , y)]