The Existence Property among Set Theories

Michael Rathjen

Department of Pure Mathematics University of Leeds

Eighth Panhellenic Logic Symposium Ioannina

July 4th 2011

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 1 Intuitionism

2 The Existence Property and other properties

3 The Existence Property and Collection

Plan of the Talk

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 2 The Existence Property and other properties

3 The Existence Property and Collection

Plan of the Talk

1 Intuitionism

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 3 The Existence Property and Collection

Plan of the Talk

1 Intuitionism

2 The Existence Property and other properties

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Plan of the Talk

1 Intuitionism

2 The Existence Property and other properties

3 The Existence Property and Collection

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES All flavors of constructivism seem to demand that: The correctness of an existential claim (∃x ∈ A)ϕ(x) is to be guaranteed by warrants from which both an object x0 ∈ A and a further warrant for ϕ(x0) are constructible.

Existentialism

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES The correctness of an existential claim (∃x ∈ A)ϕ(x) is to be guaranteed by warrants from which both an object x0 ∈ A and a further warrant for ϕ(x0) are constructible.

Existentialism

All flavors of constructivism seem to demand that:

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Existentialism

All flavors of constructivism seem to demand that: The correctness of an existential claim (∃x ∈ A)ϕ(x) is to be guaranteed by warrants from which both an object x0 ∈ A and a further warrant for ϕ(x0) are constructible.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.

Bishop: Man and God

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES If God has mathematics of his own that needs to be done, let him do it himself.

Bishop: Man and God

When a man proves a positive integer to exist, he should show how to find it.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Bishop: Man and God

When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Computational content: Witness and program extraction from proofs.

• Intuitionistically proved theorems hold in more generality: The internal logic of topoi is intuitionistic logic.

• Axiomatic Freedom Adopt axioms that are classically refutable but intuitionistically preserve algorithmic truth (E.g. All f : R → R are continuous).

Why Intuitionistic Theories?

• Philosophical Reasons: Brouwer, Dummett, Martin-Löf,..

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Intuitionistically proved theorems hold in more generality: The internal logic of topoi is intuitionistic logic.

• Axiomatic Freedom Adopt axioms that are classically refutable but intuitionistically preserve algorithmic truth (E.g. All f : R → R are continuous).

Why Intuitionistic Theories?

• Philosophical Reasons: Brouwer, Dummett, Martin-Löf,..

• Computational content: Witness and program extraction from proofs.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Axiomatic Freedom Adopt axioms that are classically refutable but intuitionistically preserve algorithmic truth (E.g. All f : R → R are continuous).

Why Intuitionistic Theories?

• Philosophical Reasons: Brouwer, Dummett, Martin-Löf,..

• Computational content: Witness and program extraction from proofs.

• Intuitionistically proved theorems hold in more generality: The internal logic of topoi is intuitionistic logic.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Why Intuitionistic Theories?

• Philosophical Reasons: Brouwer, Dummett, Martin-Löf,..

• Computational content: Witness and program extraction from proofs.

• Intuitionistically proved theorems hold in more generality: The internal logic of topoi is intuitionistic logic.

• Axiomatic Freedom Adopt axioms that are classically refutable but intuitionistically preserve algorithmic truth (E.g. All f : R → R are continuous).

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Not formalized: Brouwer 1907 (philosophical basis), 1918 (mathematical starting point) Heyting 1930: intuitionistic predicate logic and arithmetic Negative translation: Kolmogorov 1925, Gentzen and Gödel 1933.

Formalization of intuitionistic logic

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Heyting 1930: intuitionistic predicate logic and arithmetic Negative translation: Kolmogorov 1925, Gentzen and Gödel 1933.

Formalization of intuitionistic logic

Not formalized: Brouwer 1907 (philosophical basis), 1918 (mathematical starting point)

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Negative translation: Kolmogorov 1925, Gentzen and Gödel 1933.

Formalization of intuitionistic logic

Not formalized: Brouwer 1907 (philosophical basis), 1918 (mathematical starting point) Heyting 1930: intuitionistic predicate logic and arithmetic

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Formalization of intuitionistic logic

Not formalized: Brouwer 1907 (philosophical basis), 1918 (mathematical starting point) Heyting 1930: intuitionistic predicate logic and arithmetic Negative translation: Kolmogorov 1925, Gentzen and Gödel 1933.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Kleene’s 1945 realizability for HA

a realizer of has the form A atomic any e providing A is true. A ∧ B (a, b), where a is a realizer of A and b is a realizer of B. A ∨ B (0, a), where a is a realizer of A, or (1, b), where b is a realizer of B ∃x B(x)(n, b), where b is a realizer of B(n¯).

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Kleene’s 1945 realizability

a realizer of has the form A → B e, where e is the Gödel number of a Turing

machine Me such that Me halts with a realizer

for B whenever a realizer of A is run on Me. ¬A any e providing there is no realizer for A. ∀x B(x) e, where e is a Gödel number of a Turing

machine Me such that Me outputs a realizer for A(n¯) when run on n.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Basic Assumptions

Let T be a theory whose language, L(T ), encompasses the language of set theory. Moreover, for simplicity, we shall assume that L(T ) has a constant ω denoting the set of von Neumann natural numbers and for each n a constant n¯ denoting the n-th element of ω.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 1 T has the disjunction property, DP, if whenever

T ` ψ ∨ θ

holds for sentences ψ and θ of T , then

T ` ψ or T ` θ.

The Disjunction Property

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES The Disjunction Property

1 T has the disjunction property, DP, if whenever

T ` ψ ∨ θ

holds for sentences ψ and θ of T , then

T ` ψ or T ` θ.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 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

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 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.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 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)].

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Gödel (1932) observed that intuitionistic propositional logic has the DP.

• Gentzen (1934): Intuitionistic predicate logic has the DP and EP.

• Kleene (1945): HA has the DP and NEP.

• Joan Moschovakis (1965): DP, NEP and EP for (many) systems of intuitionistic analysis.

Some History

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Gentzen (1934): Intuitionistic predicate logic has the DP and EP.

• Kleene (1945): HA has the DP and NEP.

• Joan Moschovakis (1965): DP, NEP and EP for (many) systems of intuitionistic analysis.

Some History

• Gödel (1932) observed that intuitionistic propositional logic has the DP.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Kleene (1945): HA has the DP and NEP.

• Joan Moschovakis (1965): DP, NEP and EP for (many) systems of intuitionistic analysis.

Some History

• Gödel (1932) observed that intuitionistic propositional logic has the DP.

• Gentzen (1934): Intuitionistic predicate logic has the DP and EP.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Joan Moschovakis (1965): DP, NEP and EP for (many) systems of intuitionistic analysis.

Some History

• Gödel (1932) observed that intuitionistic propositional logic has the DP.

• Gentzen (1934): Intuitionistic predicate logic has the DP and EP.

• Kleene (1945): HA has the DP and NEP.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Some History

• Gödel (1932) observed that intuitionistic propositional logic has the DP.

• Gentzen (1934): Intuitionistic predicate logic has the DP and EP.

• Kleene (1945): HA has the DP and NEP.

• Joan Moschovakis (1965): DP, NEP and EP for (many) systems of intuitionistic analysis.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Ignoring the trivial counterexamples, classical theories never have the DP or the NEP.

• Z (Zermelo set theory), ZF, and ZF are known not to have the EP.

• ZFC proves that R is well-orderable, but it cannot prove that there is a definable well-ordering of R.

Remarks about Classical Set Theories

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Z (Zermelo set theory), ZF, and ZF are known not to have the EP.

• ZFC proves that R is well-orderable, but it cannot prove that there is a definable well-ordering of R.

Remarks about Classical Set Theories

• Ignoring the trivial counterexamples, classical theories never have the DP or the NEP.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • ZFC proves that R is well-orderable, but it cannot prove that there is a definable well-ordering of R.

Remarks about Classical Set Theories

• Ignoring the trivial counterexamples, classical theories never have the DP or the NEP.

• Z (Zermelo set theory), ZF, and ZF are known not to have the EP.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Remarks about Classical Set Theories

• Ignoring the trivial counterexamples, classical theories never have the DP or the NEP.

• Z (Zermelo set theory), ZF, and ZF are known not to have the EP.

• ZFC proves that R is well-orderable, but it cannot prove that there is a definable well-ordering of R.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 1 1 (Kondo, Addison) If ZF ` ∃x ∈ R ϕ(x) and ϕ(x) is Σ2, then ZF ` ∃!x ∈ R ϑ(x) and ZF ` ∃x ∈ R [ϑ(x) ∧ ϕ(x)] for some 1 Σ2 formula ϑ.

2 1 (Feferman, Lévy) EP fails for Π2 in ZF and ZFC.

3 (Y. Moschovakis) ZF + Projective Determinacy has the projective existence property (ϕ(x), ϑ(x) projective).

• Nevertheless, fragments of the EP, known as uniformization properties, sometimes hold.

Remarks about partial EP

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 1 1 (Kondo, Addison) If ZF ` ∃x ∈ R ϕ(x) and ϕ(x) is Σ2, then ZF ` ∃!x ∈ R ϑ(x) and ZF ` ∃x ∈ R [ϑ(x) ∧ ϕ(x)] for some 1 Σ2 formula ϑ.

2 1 (Feferman, Lévy) EP fails for Π2 in ZF and ZFC.

3 (Y. Moschovakis) ZF + Projective Determinacy has the projective existence property (ϕ(x), ϑ(x) projective).

Remarks about partial EP

• Nevertheless, fragments of the EP, known as uniformization properties, sometimes hold.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 2 1 (Feferman, Lévy) EP fails for Π2 in ZF and ZFC.

3 (Y. Moschovakis) ZF + Projective Determinacy has the projective existence property (ϕ(x), ϑ(x) projective).

Remarks about partial EP

• Nevertheless, fragments of the EP, known as uniformization properties, sometimes hold.

1 1 (Kondo, Addison) If ZF ` ∃x ∈ R ϕ(x) and ϕ(x) is Σ2, then ZF ` ∃!x ∈ R ϑ(x) and ZF ` ∃x ∈ R [ϑ(x) ∧ ϕ(x)] for some 1 Σ2 formula ϑ.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 3 (Y. Moschovakis) ZF + Projective Determinacy has the projective existence property (ϕ(x), ϑ(x) projective).

Remarks about partial EP

• Nevertheless, fragments of the EP, known as uniformization properties, sometimes hold.

1 1 (Kondo, Addison) If ZF ` ∃x ∈ R ϕ(x) and ϕ(x) is Σ2, then ZF ` ∃!x ∈ R ϑ(x) and ZF ` ∃x ∈ R [ϑ(x) ∧ ϕ(x)] for some 1 Σ2 formula ϑ.

2 1 (Feferman, Lévy) EP fails for Π2 in ZF and ZFC.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Remarks about partial EP

• Nevertheless, fragments of the EP, known as uniformization properties, sometimes hold.

1 1 (Kondo, Addison) If ZF ` ∃x ∈ R ϕ(x) and ϕ(x) is Σ2, then ZF ` ∃!x ∈ R ϑ(x) and ZF ` ∃x ∈ R [ϑ(x) ∧ ϕ(x)] for some 1 Σ2 formula ϑ.

2 1 (Feferman, Lévy) EP fails for Π2 in ZF and ZFC.

3 (Y. Moschovakis) ZF + Projective Determinacy has the projective existence property (ϕ(x), ϑ(x) projective).

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Reasonable classical set theories can have the full EP.

Theorem An extension T of ZF has the EP if and only if T proves that all sets are ordinal definable, i.e., T ` V = OD.

Classical theories and EP

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Theorem An extension T of ZF has the EP if and only if T proves that all sets are ordinal definable, i.e., T ` V = OD.

Classical theories and EP

• Reasonable classical set theories can have the full EP.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Classical theories and EP

• Reasonable classical set theories can have the full EP.

Theorem An extension T of ZF has the EP if and only if T proves that all sets are ordinal definable, i.e., T ` V = OD.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES CST based on intuitionistic logic Many sorted system: numbers, sets, functions Axioms (simplified) * Extensionality • Pairing, Union, Infinity (or N is a set) • Bounded Separation • Exponentiation: A, B sets ⇒ AB set. • Replacement

Myhill’s Constructive set theory 1975

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Many sorted system: numbers, sets, functions Axioms (simplified) * Extensionality • Pairing, Union, Infinity (or N is a set) • Bounded Separation • Exponentiation: A, B sets ⇒ AB set. • Replacement

Myhill’s Constructive set theory 1975

CST based on intuitionistic logic

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Axioms (simplified) * Extensionality • Pairing, Union, Infinity (or N is a set) • Bounded Separation • Exponentiation: A, B sets ⇒ AB set. • Replacement

Myhill’s Constructive set theory 1975

CST based on intuitionistic logic Many sorted system: numbers, sets, functions

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES * Extensionality • Pairing, Union, Infinity (or N is a set) • Bounded Separation • Exponentiation: A, B sets ⇒ AB set. • Replacement

Myhill’s Constructive set theory 1975

CST based on intuitionistic logic Many sorted system: numbers, sets, functions Axioms (simplified)

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Pairing, Union, Infinity (or N is a set) • Bounded Separation • Exponentiation: A, B sets ⇒ AB set. • Replacement

Myhill’s Constructive set theory 1975

CST based on intuitionistic logic Many sorted system: numbers, sets, functions Axioms (simplified) * Extensionality

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Bounded Separation • Exponentiation: A, B sets ⇒ AB set. • Replacement

Myhill’s Constructive set theory 1975

CST based on intuitionistic logic Many sorted system: numbers, sets, functions Axioms (simplified) * Extensionality • Pairing, Union, Infinity (or N is a set)

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Exponentiation: A, B sets ⇒ AB set. • Replacement

Myhill’s Constructive set theory 1975

CST based on intuitionistic logic Many sorted system: numbers, sets, functions Axioms (simplified) * Extensionality • Pairing, Union, Infinity (or N is a set) • Bounded Separation

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Replacement

Myhill’s Constructive set theory 1975

CST based on intuitionistic logic Many sorted system: numbers, sets, functions Axioms (simplified) * Extensionality • Pairing, Union, Infinity (or N is a set) • Bounded Separation • Exponentiation: A, B sets ⇒ AB set.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Myhill’s Constructive set theory 1975

CST based on intuitionistic logic Many sorted system: numbers, sets, functions Axioms (simplified) * Extensionality • Pairing, Union, Infinity (or N is a set) • Bounded Separation • Exponentiation: A, B sets ⇒ AB set. • Replacement

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES * 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),

Myhill’s IZFR: IZF with Replacement instead of Collection

Intuitionistic Zermelo-Fraenkel set theory, IZF

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • 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),

Myhill’s IZFR: IZF with Replacement instead of Collection

Intuitionistic Zermelo-Fraenkel set theory, IZF

* Extensionality

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • 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),

Myhill’s IZFR: IZF with Replacement instead of Collection

Intuitionistic Zermelo-Fraenkel set theory, IZF

* Extensionality • Pairing, Union, Infinity

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES # 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),

Myhill’s IZFR: IZF with Replacement instead of Collection

Intuitionistic Zermelo-Fraenkel set theory, IZF

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

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES * Set Induction

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

Myhill’s IZFR: IZF with Replacement instead of Collection

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)

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Myhill’s IZFR: IZF with Replacement instead of Collection

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),

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 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),

Myhill’s IZFR: IZF with Replacement instead of Collection

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES * 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

Constructive Zermelo-Fraenkel set theory, CZF

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • 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

Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • 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

Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality • Pairing, Union, Infinity

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • 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

Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality • Pairing, Union, Infinity • Bounded Separation

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES # 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

Constructive Zermelo-Fraenkel set theory, CZF

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

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES * Set Induction scheme

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)]

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 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

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Explicit set existence axioms: e.g. Separation, Replacement, Exponentiation • Non-explicit set existence axioms: e.g. in classical set theory Axioms of Choice • Non-explicit set existence axioms in intuitionistic set theory: e.g. Axioms of Choice, (Strong) Collection, Subset Collection, Regular Extension Axiom

Two types of set existence axioms

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Non-explicit set existence axioms: e.g. in classical set theory Axioms of Choice • Non-explicit set existence axioms in intuitionistic set theory: e.g. Axioms of Choice, (Strong) Collection, Subset Collection, Regular Extension Axiom

Two types of set existence axioms

• Explicit set existence axioms: e.g. Separation, Replacement, Exponentiation

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Non-explicit set existence axioms in intuitionistic set theory: e.g. Axioms of Choice, (Strong) Collection, Subset Collection, Regular Extension Axiom

Two types of set existence axioms

• Explicit set existence axioms: e.g. Separation, Replacement, Exponentiation • Non-explicit set existence axioms: e.g. in classical set theory Axioms of Choice

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Two types of set existence axioms

• Explicit set existence axioms: e.g. Separation, Replacement, Exponentiation • Non-explicit set existence axioms: e.g. in classical set theory Axioms of Choice • Non-explicit set existence axioms in intuitionistic set theory: e.g. Axioms of Choice, (Strong) Collection, Subset Collection, Regular Extension Axiom

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Theorem 1. (Myhill) IZFR and CST have the DP, NEP, and the EP.

Theorem 2. (Beeson) IZF has the DP and the NEP.

Theorem 3. (Friedman, Scedrov) IZF does not have the EP.

Some History

Let IZFR result from IZF by replacing Collection with Replacement, and let CST be Myhill’s constructive set theory.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Theorem 2. (Beeson) IZF has the DP and the NEP.

Theorem 3. (Friedman, Scedrov) IZF does not have the EP.

Some History

Let IZFR result from IZF by replacing Collection with Replacement, and let CST be Myhill’s constructive set theory. Theorem 1. (Myhill) IZFR and CST have the DP, NEP, and the EP.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Theorem 3. (Friedman, Scedrov) IZF does not have the EP.

Some History

Let IZFR result from IZF by replacing Collection with Replacement, and let CST be Myhill’s constructive set theory. Theorem 1. (Myhill) IZFR and CST have the DP, NEP, and the EP.

Theorem 2. (Beeson) IZF has the DP and the NEP.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Some History

Let IZFR result from IZF by replacing Collection with Replacement, and let CST be Myhill’s constructive set theory. Theorem 1. (Myhill) IZFR and CST have the DP, NEP, and the EP.

Theorem 2. (Beeson) IZF has the DP and the NEP.

Theorem 3. (Friedman, Scedrov) IZF does not have the EP.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Realizability Theorem

Realizability with truth. Theorem: (R) For every theorem θ of CZF, there exists an application term s such that

CZF ` (s t θ). Moreover, the proof of this soundness theorem is effective in that the application term s can be effectively constructed from the CZF proof of θ.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Theorem: (R) The DP and the NEP hold true for CZF, CZF + REA and CZF + Large Set Axioms. One can also add Subset Collection and the following choice principles:

ACω, DC, RDC, PAx.

Theorem: The DP and the NEP hold true for IZF, IZF + REA and IZF + Large Set Axioms. One can also add ACω, DC, RDC, PAx.

The Main Theorem

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Theorem: The DP and the NEP hold true for IZF, IZF + REA and IZF + Large Set Axioms. One can also add ACω, DC, RDC, PAx.

The Main Theorem

Theorem: (R) The DP and the NEP hold true for CZF, CZF + REA and CZF + Large Set Axioms. One can also add Subset Collection and the following choice principles:

ACω, DC, RDC, PAx.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES The Main Theorem

Theorem: (R) The DP and the NEP hold true for CZF, CZF + REA and CZF + Large Set Axioms. One can also add Subset Collection and the following choice principles:

ACω, DC, RDC, PAx.

Theorem: The DP and the NEP hold true for IZF, IZF + REA and IZF + Large Set Axioms. One can also add ACω, DC, RDC, PAx.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 1 Adding Powerset or other axioms to CZF doesn’t change the results.

2 It can be adapted to other PCAs, e.g. the second Kleene algebra to show that provable functions on Baire space are continuous.

1 R.: The disjunction and related properties for constructive Zermelo-Fraenkel set theory. Journal of Symbolic Logic 70 (2005) 1233–1254.

2 R.: Metamathematical Properties of Intuitionistic Set Theories with Choice Principles. In: S. B. Cooper, B. Löwe, A. Sorbi (eds.): New Computational Paradigms: Changing Conceptions of What is Computable (Springer, New York, 2008) 287–312.

• This notion of realizability is very robust.

References:

Remarks

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 1 R.: The disjunction and related properties for constructive Zermelo-Fraenkel set theory. Journal of Symbolic Logic 70 (2005) 1233–1254.

2 R.: Metamathematical Properties of Intuitionistic Set Theories with Choice Principles. In: S. B. Cooper, B. Löwe, A. Sorbi (eds.): New Computational Paradigms: Changing Conceptions of What is Computable (Springer, New York, 2008) 287–312.

1 Adding Powerset or other axioms to CZF doesn’t change the results.

2 It can be adapted to other PCAs, e.g. the second Kleene algebra to show that provable functions on Baire space are continuous. References:

Remarks

• This notion of realizability is very robust.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 1 R.: The disjunction and related properties for constructive Zermelo-Fraenkel set theory. Journal of Symbolic Logic 70 (2005) 1233–1254.

2 R.: Metamathematical Properties of Intuitionistic Set Theories with Choice Principles. In: S. B. Cooper, B. Löwe, A. Sorbi (eds.): New Computational Paradigms: Changing Conceptions of What is Computable (Springer, New York, 2008) 287–312.

2 It can be adapted to other PCAs, e.g. the second Kleene algebra to show that provable functions on Baire space are continuous. References:

Remarks

• This notion of realizability is very robust. 1 Adding Powerset or other axioms to CZF doesn’t change the results.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 1 R.: The disjunction and related properties for constructive Zermelo-Fraenkel set theory. Journal of Symbolic Logic 70 (2005) 1233–1254.

2 R.: Metamathematical Properties of Intuitionistic Set Theories with Choice Principles. In: S. B. Cooper, B. Löwe, A. Sorbi (eds.): New Computational Paradigms: Changing Conceptions of What is Computable (Springer, New York, 2008) 287–312.

References:

Remarks

• This notion of realizability is very robust. 1 Adding Powerset or other axioms to CZF doesn’t change the results.

2 It can be adapted to other PCAs, e.g. the second Kleene algebra to show that provable functions on Baire space are continuous.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 1 R.: The disjunction and related properties for constructive Zermelo-Fraenkel set theory. Journal of Symbolic Logic 70 (2005) 1233–1254.

2 R.: Metamathematical Properties of Intuitionistic Set Theories with Choice Principles. In: S. B. Cooper, B. Löwe, A. Sorbi (eds.): New Computational Paradigms: Changing Conceptions of What is Computable (Springer, New York, 2008) 287–312.

Remarks

• This notion of realizability is very robust. 1 Adding Powerset or other axioms to CZF doesn’t change the results.

2 It can be adapted to other PCAs, e.g. the second Kleene algebra to show that provable functions on Baire space are continuous. References:

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 2 R.: Metamathematical Properties of Intuitionistic Set Theories with Choice Principles. In: S. B. Cooper, B. Löwe, A. Sorbi (eds.): New Computational Paradigms: Changing Conceptions of What is Computable (Springer, New York, 2008) 287–312.

Remarks

• This notion of realizability is very robust. 1 Adding Powerset or other axioms to CZF doesn’t change the results.

2 It can be adapted to other PCAs, e.g. the second Kleene algebra to show that provable functions on Baire space are continuous. References: 1 R.: The disjunction and related properties for constructive Zermelo-Fraenkel set theory. Journal of Symbolic Logic 70 (2005) 1233–1254.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Remarks

• This notion of realizability is very robust. 1 Adding Powerset or other axioms to CZF doesn’t change the results.

2 It can be adapted to other PCAs, e.g. the second Kleene algebra to show that provable functions on Baire space are continuous. References: 1 R.: The disjunction and related properties for constructive Zermelo-Fraenkel set theory. Journal of Symbolic Logic 70 (2005) 1233–1254.

2 R.: Metamathematical Properties of Intuitionistic Set Theories with Choice Principles. In: S. B. Cooper, B. Löwe, A. Sorbi (eds.): New Computational Paradigms: Changing Conceptions of What is Computable (Springer, New York, 2008) 287–312.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Collection is ∀x ∈ a ∃y A(x, y) → ∃b ∀x ∈ a ∃y ∈ b A(x, y). This is in IZF equivalent to ∃b [∀x ∈ a ∃y A(x, y) → ∀x ∈ a ∃y ∈ b A(x, y)] • Let B(z) be a formula expressing that z is an uncountable cardinal. Let B∗(z) result from B(z) by replacing every atomic subformula D of B(z) by D ∨ ∀uv(u ∈ v ∨ ¬ u ∈ v ). EP fails for IZF for the following instance: ∃y [∀x ∈ 1 ∃z B∗(z) → ∀x ∈ 1 ∃z ∈ y B∗(z)] .

Failure of EP for IZF

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES This is in IZF equivalent to ∃b [∀x ∈ a ∃y A(x, y) → ∀x ∈ a ∃y ∈ b A(x, y)] • Let B(z) be a formula expressing that z is an uncountable cardinal. Let B∗(z) result from B(z) by replacing every atomic subformula D of B(z) by D ∨ ∀uv(u ∈ v ∨ ¬ u ∈ v ). EP fails for IZF for the following instance: ∃y [∀x ∈ 1 ∃z B∗(z) → ∀x ∈ 1 ∃z ∈ y B∗(z)] .

Failure of EP for IZF

Collection is ∀x ∈ a ∃y A(x, y) → ∃b ∀x ∈ a ∃y ∈ b A(x, y).

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Let B(z) be a formula expressing that z is an uncountable cardinal. Let B∗(z) result from B(z) by replacing every atomic subformula D of B(z) by D ∨ ∀uv(u ∈ v ∨ ¬ u ∈ v ). EP fails for IZF for the following instance: ∃y [∀x ∈ 1 ∃z B∗(z) → ∀x ∈ 1 ∃z ∈ y B∗(z)] .

Failure of EP for IZF

Collection is ∀x ∈ a ∃y A(x, y) → ∃b ∀x ∈ a ∃y ∈ b A(x, y). This is in IZF equivalent to ∃b [∀x ∈ a ∃y A(x, y) → ∀x ∈ a ∃y ∈ b A(x, y)]

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES EP fails for IZF for the following instance: ∃y [∀x ∈ 1 ∃z B∗(z) → ∀x ∈ 1 ∃z ∈ y B∗(z)] .

Failure of EP for IZF

Collection is ∀x ∈ a ∃y A(x, y) → ∃b ∀x ∈ a ∃y ∈ b A(x, y). This is in IZF equivalent to ∃b [∀x ∈ a ∃y A(x, y) → ∀x ∈ a ∃y ∈ b A(x, y)] • Let B(z) be a formula expressing that z is an uncountable cardinal. Let B∗(z) result from B(z) by replacing every atomic subformula D of B(z) by D ∨ ∀uv(u ∈ v ∨ ¬ u ∈ v ).

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Failure of EP for IZF

Collection is ∀x ∈ a ∃y A(x, y) → ∃b ∀x ∈ a ∃y ∈ b A(x, y). This is in IZF equivalent to ∃b [∀x ∈ a ∃y A(x, y) → ∀x ∈ a ∃y ∈ b A(x, y)] • Let B(z) be a formula expressing that z is an uncountable cardinal. Let B∗(z) result from B(z) by replacing every atomic subformula D of B(z) by D ∨ ∀uv(u ∈ v ∨ ¬ u ∈ v ). EP fails for IZF for the following instance: ∃y [∀x ∈ 1 ∃z B∗(z) → ∀x ∈ 1 ∃z ∈ y B∗(z)] .

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • (Beeson 1985) Does any reasonable set theory with Collection have the existential definability property?

Problems

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Problems

• (Beeson 1985) Does any reasonable set theory with Collection have the existential definability property?

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 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

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 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)].

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • We would like to have unlimited application of sets to sets, i.e. we would like to assign a meaning to the symbol {a}(x) where a and x are sets. • Known as E-recursion or set recursion • However, we shall introduce an extended notion of E-computability, christened E℘-computability, rendering the function exp(a, b) = ab computable as well. • Classically, E℘-computability is related to power recursion, where the power set operation is regarded to be an initial function. Notion studied by Yiannis Moschovakis and Larry Moss.

Extended E-recursive functions

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Known as E-recursion or set recursion • However, we shall introduce an extended notion of E-computability, christened E℘-computability, rendering the function exp(a, b) = ab computable as well. • Classically, E℘-computability is related to power recursion, where the power set operation is regarded to be an initial function. Notion studied by Yiannis Moschovakis and Larry Moss.

Extended E-recursive functions

• We would like to have unlimited application of sets to sets, i.e. we would like to assign a meaning to the symbol {a}(x) where a and x are sets.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • However, we shall introduce an extended notion of E-computability, christened E℘-computability, rendering the function exp(a, b) = ab computable as well. • Classically, E℘-computability is related to power recursion, where the power set operation is regarded to be an initial function. Notion studied by Yiannis Moschovakis and Larry Moss.

Extended E-recursive functions

• We would like to have unlimited application of sets to sets, i.e. we would like to assign a meaning to the symbol {a}(x) where a and x are sets. • Known as E-recursion or set recursion

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • Classically, E℘-computability is related to power recursion, where the power set operation is regarded to be an initial function. Notion studied by Yiannis Moschovakis and Larry Moss.

Extended E-recursive functions

• We would like to have unlimited application of sets to sets, i.e. we would like to assign a meaning to the symbol {a}(x) where a and x are sets. • Known as E-recursion or set recursion • However, we shall introduce an extended notion of E-computability, christened E℘-computability, rendering the function exp(a, b) = ab computable as well.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Extended E-recursive functions

• We would like to have unlimited application of sets to sets, i.e. we would like to assign a meaning to the symbol {a}(x) where a and x are sets. • Known as E-recursion or set recursion • However, we shall introduce an extended notion of E-computability, christened E℘-computability, rendering the function exp(a, b) = ab computable as well. • Classically, E℘-computability is related to power recursion, where the power set operation is regarded to be an initial function. Notion studied by Yiannis Moschovakis and Larry Moss.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Realizability with sets of witnesses

We use the expression a 6= ∅ to convey the positive fact that the set a is inhabited, that is ∃x x ∈ a. We define a relation a wt B between sets and set-theoretic formulae.

a • f wt B will be an abbreviation for

∃x[a • f ' x ∧ x wt B]

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES a wt A iff A holds true, whenever A is an atomic formula

a wt A ∧ B iff 0a wt A ∧ 1a wt B   a wt A ∨ B iff a 6= ∅ ∧ (∀d ∈ a)( 0d = 0 ∧ 1d wt A ∨   0d = 1 ∧ 1d wt B )

a wt ¬A iff ¬A ∧ ∀c ¬c wt A   a wt A → B iff (A → B) ∧ ∀c c wt A → a • c wt B a wt (∀x ∈ b) A iff (∀c ∈ b) a • c wt A[x/c] a wt (∃x ∈ b) A iff a 6= ∅ ∧ (∀d ∈ a)[0d ∈ b ∧ 1d wt A[x/0d]

a wt ∀xA iff ∀c a • c wt A[x/c]

a wt ∃xA iff a 6= ∅ ∧ (∀d ∈ a) 1d wt A[x/0d]

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES a wt A iff A holds true, whenever A is an atomic formula

a wt A ∧ B iff 0a wt A ∧ 1a wt B   a wt A ∨ B iff a 6= ∅ ∧ (∀d ∈ a)( 0d = 0 ∧ 1d wt A ∨   0d = 1 ∧ 1d wt B )

a wt ¬A iff ¬A ∧ ∀c ¬c wt A   a wt A → B iff (A → B) ∧ ∀c c wt A → a • c wt B a wt (∀x ∈ b) A iff (∀c ∈ b) a • c wt A[x/c] a wt (∃x ∈ b) A iff a 6= ∅ ∧ (∀d ∈ a)( 0d ∈ b ∧ 1d wt A[x/0d])

a wt ∀xA iff ∀c a • c wt A[x/c]

a wt ∃xA iff a 6= ∅ ∧ (∀d ∈ a) 1d wt A[x/0d]

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Corollary

(i) CZF ` ( wt B) → B.

℘ (ii) CZF + Pow ` ( wt B) → B.

wt B iff ∃a a wt B.

If we use indices of E℘-recursive functions rather than

Eexp -recursive functions, we notate the corresponding notion of ℘ realizability by a wt B.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES (i) CZF ` ( wt B) → B.

℘ (ii) CZF + Pow ` ( wt B) → B.

wt B iff ∃a a wt B.

If we use indices of E℘-recursive functions rather than

Eexp -recursive functions, we notate the corresponding notion of ℘ realizability by a wt B. Corollary

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES ℘ (ii) CZF + Pow ` ( wt B) → B.

wt B iff ∃a a wt B.

If we use indices of E℘-recursive functions rather than

Eexp -recursive functions, we notate the corresponding notion of ℘ realizability by a wt B. Corollary

(i) CZF ` ( wt B) → B.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES wt B iff ∃a a wt B.

If we use indices of E℘-recursive functions rather than

Eexp -recursive functions, we notate the corresponding notion of ℘ realizability by a wt B. Corollary

(i) CZF ` ( wt B) → B.

℘ (ii) CZF + Pow ` ( wt B) → B.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Theorem CZF and CZF + Pow both have the weak existence property. Indeed, they both satisfy the stronger property wEP0.

A variant of wEP, dubbed wEP’, is the following: 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 EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES A variant of wEP, dubbed wEP’, is the following: 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)].

Theorem CZF and CZF + Pow both have the weak existence property. Indeed, they both satisfy the stronger property wEP0.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • 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

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 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)

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES • 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

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES 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)

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES THEOREM CZF is conservative over IKP(E) for ΣE sentences.

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

Conservativity

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES THEOREM CZF + Pow is conservative over IKP(P) for ΣP sentences.

Conservativity

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

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Conservativity

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

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

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Theorem 1: CZF has the existence property.

Theorem 2: CZF + Pow has the existence property.

Conjecture 3: CZF + Subset Collection does not have the weak existence property.

Theorems and a Conjecture

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Theorem 2: CZF + Pow has the existence property.

Conjecture 3: CZF + Subset Collection does not have the weak existence property.

Theorems and a Conjecture

Theorem 1: CZF has the existence property.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Conjecture 3: CZF + Subset Collection does not have the weak existence property.

Theorems and a Conjecture

Theorem 1: CZF has the existence property.

Theorem 2: CZF + Pow has the existence property.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES Theorems and a Conjecture

Theorem 1: CZF has the existence property.

Theorem 2: CZF + Pow has the existence property.

Conjecture 3: CZF + Subset Collection does not have the weak existence property.

THE EXISTENCE PROPERTY AMONG SET THEORIES THE EXISTENCE PROPERTY AMONG SET THEORIES