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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
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
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
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
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
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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).
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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).
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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).
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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).
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
The Bachmann-Howard ordinal
Theorem 2 The following theories have theBachmann-Howard ordinal,
ψΩ1
(εΩ1+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
The Bachmann-Howard ordinal
Theorem 2 The following theories have theBachmann-Howard ordinal,
ψΩ1
(εΩ1+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
The Bachmann-Howard ordinal
Theorem 2 The following theories have theBachmann-Howard ordinal,
ψΩ1
(εΩ1+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
The Bachmann-Howard ordinal
Theorem 2 The following theories have theBachmann-Howard ordinal,
ψΩ1
(εΩ1+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
The Bachmann-Howard ordinal
Theorem 2 The following theories have theBachmann-Howard ordinal,
ψΩ1
(εΩ1+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
The Bachmann-Howard ordinal
Theorem 2 The following theories have theBachmann-Howard ordinal,
ψΩ1
(εΩ1+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
The Bachmann-Howard ordinal
Theorem 2 The following theories have theBachmann-Howard ordinal,
ψΩ1
(εΩ1+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 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).
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 =⋃
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
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
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 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).
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 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).
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 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).
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 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).
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
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
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
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.
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-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.
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-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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Warning
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 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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Warning
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 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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Warning
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 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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Same strength
Theorem:(R. 2012) The following theories have the same proof-theoreticstrength
(i) KP(P)
(ii) CZF + Powerset
(Basically IZF with Bounded Separation)
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Same strength
Theorem:(R. 2012) The following theories have the same proof-theoreticstrength
(i) KP(P)
(ii) CZF + Powerset
(Basically IZF with Bounded Separation)
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Same strength
Theorem:(R. 2012) The following theories have the same proof-theoreticstrength
(i) KP(P)
(ii) CZF + Powerset
(Basically IZF with Bounded Separation)
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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.
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 samestrength.
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 samestrength.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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)]
(b) ∀x∀y∀z [R(x , y) ∧ R(x , z)→ y = z]
(c) Extend schemata of KP(P) to new language.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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)]
(b) ∀x∀y∀z [R(x , y) ∧ R(x , z)→ y = z]
(c) Extend schemata of KP(P) to new language.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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)]
(b) ∀x∀y∀z [R(x , y) ∧ R(x , z)→ y = z]
(c) Extend schemata of KP(P) to new language.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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)]
(b) ∀x∀y∀z [R(x , y) ∧ R(x , z)→ y = z]
(c) Extend schemata of KP(P) to new language.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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)]
(b) ∀x∀y∀z [R(x , y) ∧ R(x , z)→ y = z]
(c) Extend schemata of KP(P) to new language.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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)]
(b) ∀x∀y∀z [R(x , y) ∧ R(x , z)→ y = z]
(c) Extend schemata of KP(P) to new language.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
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)]
(b) ∀x∀y∀z [R(x , y) ∧ R(x , z)→ y = z]
(c) Extend schemata of KP(P) to new language.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Theorem:
IfKP(P) + GAC ` θ
where θ is a ΣP -sentence, then one can explicitly find anordinal (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
proves a Π14 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:
If Φ is Π14 sentence such that
KP(P) + GAC ` Φ
thenKP(P) ` Φ.
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
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
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
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
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
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
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
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
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
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
The Existence Property
1 T has the numerical existence property, NEP, ifwhenever
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
The Existence Property
1 T has the numerical existence property, NEP, ifwhenever
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
The Existence Property
1 T has the numerical existence property, NEP, ifwhenever
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Problems
Theorem. ( Friedman, Šcedrov 1985) IZF does not havethe existence property.
• (Beeson 1985) Does any reasonable set theory withCollection have the existential definability property?
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Problems
Theorem. ( Friedman, Šcedrov 1985) IZF does not havethe existence property.
• (Beeson 1985) Does any reasonable set theory withCollection have the existential definability property?
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Problems
Theorem. ( Friedman, Šcedrov 1985) IZF does not havethe existence property.
• (Beeson 1985) Does any reasonable set theory withCollection have the existential definability 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
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
IZF and wEP
Theorem IZF does not have the weak existence propertyproperty.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
IZF and wEP
Theorem IZF does not have the weak existence propertyproperty.
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, ifthe following holds: if
T ` ∀u ∃xA(u, x)
holds for a formula A(u, x) having at most the freevariables u, x , then there is a formula B(u, x) with exactlyu, 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
The Uniform Weak Existence Property
T has the uniform weak existence property, uwEP, ifthe following holds: if
T ` ∀u ∃xA(u, x)
holds for a formula A(u, x) having at most the freevariables u, x , then there is a formula B(u, x) with exactlyu, 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 havethe uniform weak existence property.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Theorem The theories CZF−, CZF and CZF + Pow havethe uniform weak existence property.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Even better
• THEOREM IfCZF ` ∃x A(x)
then one can effectively construct a ΣE formula B(y) suchthat
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
Even better
• THEOREM IfCZF ` ∃x A(x)
then one can effectively construct a ΣE formula B(y) suchthat
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
Even better
• THEOREM IfCZF + Pow ` ∃x A(x)
then one can effectively construct a ΣP formula B(y) suchthat
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
Even better
• THEOREM IfCZF + Pow ` ∃x A(x)
then one can effectively construct a ΣP formula B(y) suchthat
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
Intuitionistic Power Kripke-Platek Set Theory and friends
IKP(P) is intuitionistic Power Kripke-Platek Set Theory.
We call a formula of L∈ ∆E0 if all its quantifiers are of theform 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,∆E0 -Separation and ∆E0 -Collection.
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.
We call a formula of L∈ ∆E0 if all its quantifiers are of theform 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,∆E0 -Separation and ∆E0 -Collection.
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.
We call a formula of L∈ ∆E0 if all its quantifiers are of theform 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,∆E0 -Separation and ∆E0 -Collection.
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.
We call a formula of L∈ ∆E0 if all its quantifiers are of theform 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,∆E0 -Separation and ∆E0 -Collection.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Conservativity
THEOREMCZF is conservative over IKP(E) for ΣE sentences.
THEOREMCZF + Pow is conservative over IKP(P) for ΣP sentences.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Conservativity
THEOREMCZF is conservative over IKP(E) for ΣE sentences.
THEOREMCZF + Pow is conservative over IKP(P) for ΣP sentences.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Conservativity
THEOREMCZF is conservative over IKP(E) for ΣE sentences.
THEOREMCZF + Pow is conservative over IKP(P) for ΣP sentences.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Existence property I
Theorem 1: IKP has the existence property for Σ1formulae.
Theorem 2: IKP(E) has the existence property for ΣE1formulae
Theorem 3: IKP(P) has the existence property for ΣP1formulae.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Existence property I
Theorem 1: IKP has the existence property for Σ1formulae.
Theorem 2: IKP(E) has the existence property for ΣE1formulae
Theorem 3: IKP(P) has the existence property for ΣP1formulae.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Existence property I
Theorem 1: IKP has the existence property for Σ1formulae.
Theorem 2: IKP(E) has the existence property for ΣE1formulae
Theorem 3: IKP(P) has the existence property for ΣP1formulae.
POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE POWER KP, ORDINAL ANALYSIS, GLOBAL CHOICE
Existence property I
Theorem 1: IKP has the existence property for Σ1formulae.
Theorem 2: IKP(E) has the existence property for ΣE1formulae
Theorem 3: IKP(P) has the existence property for ΣP1formulae.
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 nothave the weak 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 nothave the weak 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 nothave the weak 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 nothave the weak 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 nothave the weak existence property.
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
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 existenceproperty, 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 Philosophyand Foundations of Mathematics in Honour of PerMartin-Löf, (Springer, Dordrecht, Heidelberg, 2012)313–349.
3 M. Rathjen: Relativized ordinal analysis: The case ofPower Kripke-Platek set theory. Annals of Pure andApplied 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
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