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The Existence Property among Set Theories Michael Rathjen Department of Pure Mathematics University of Leeds Eighth Panhellenic Logic Symposium Ioannina July 4 th 2011 T HE E XISTENCE P ROPERTY AMONG S ET T HEORIES T HE E XISTENCE P ROPERTY

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SLIDE 1

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

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SLIDE 2

Plan of the Talk

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

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SLIDE 3

Plan of the Talk

1

Intuitionism

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

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SLIDE 4

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

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SLIDE 5

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

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SLIDE 6

Existentialism

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

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SLIDE 7

Existentialism

All flavors of constructivism seem to demand that:

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

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SLIDE 8

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

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SLIDE 9

Bishop: Man and God

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

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SLIDE 10

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

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SLIDE 11

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

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SLIDE 12

Why Intuitionistic Theories?

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

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

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SLIDE 13

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

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SLIDE 14

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

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SLIDE 15

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

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SLIDE 16

Formalization of intuitionistic logic

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

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SLIDE 17

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

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SLIDE 18

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

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SLIDE 19

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

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SLIDE 20

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,

  • r (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

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SLIDE 21

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

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SLIDE 22

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

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SLIDE 23

The Disjunction Property

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SLIDE 24

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

  • r T ⊢ θ.

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SLIDE 25

The Existence Property

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

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SLIDE 26

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

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SLIDE 27

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

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SLIDE 28

Some History

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

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SLIDE 29

Some History

  • Gödel (1932) observed that intuitionistic propositional logic

has the DP.

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

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SLIDE 30

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

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SLIDE 31

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

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SLIDE 32

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

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SLIDE 33

Remarks about Classical Set Theories

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

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SLIDE 34

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

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SLIDE 35

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

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SLIDE 36

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

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SLIDE 37

Remarks about partial EP

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

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SLIDE 38

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

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SLIDE 39

Remarks about partial EP

  • Nevertheless, fragments of the EP, known as

uniformization properties, sometimes hold.

1

(Kondo, Addison) If ZF ⊢ ∃x ∈ R ϕ(x) and ϕ(x) is Σ1

2, then

ZF ⊢ ∃!x ∈ R ϑ(x) and ZF ⊢ ∃x ∈ R [ϑ(x) ∧ ϕ(x)] for some Σ1

2 formula ϑ.

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

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SLIDE 40

Remarks about partial EP

  • Nevertheless, fragments of the EP, known as

uniformization properties, sometimes hold.

1

(Kondo, Addison) If ZF ⊢ ∃x ∈ R ϕ(x) and ϕ(x) is Σ1

2, then

ZF ⊢ ∃!x ∈ R ϑ(x) and ZF ⊢ ∃x ∈ R [ϑ(x) ∧ ϕ(x)] for some Σ1

2 formula ϑ.

2

(Feferman, Lévy) EP fails for Π1

2 in ZF and ZFC.

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

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SLIDE 41

Remarks about partial EP

  • Nevertheless, fragments of the EP, known as

uniformization properties, sometimes hold.

1

(Kondo, Addison) If ZF ⊢ ∃x ∈ R ϕ(x) and ϕ(x) is Σ1

2, then

ZF ⊢ ∃!x ∈ R ϑ(x) and ZF ⊢ ∃x ∈ R [ϑ(x) ∧ ϕ(x)] for some Σ1

2 formula ϑ.

2

(Feferman, Lévy) EP fails for Π1

2 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

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SLIDE 42

Classical theories and EP

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

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SLIDE 43

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

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SLIDE 44

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.

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SLIDE 45

Myhill’s Constructive set theory 1975

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

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SLIDE 46

Myhill’s Constructive set theory 1975

CST based on intuitionistic logic

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

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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

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SLIDE 48

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

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SLIDE 49

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

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SLIDE 50

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

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SLIDE 51

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

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SLIDE 52

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

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SLIDE 53

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

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SLIDE 54

Intuitionistic Zermelo-Fraenkel set theory, IZF

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

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SLIDE 55

Intuitionistic Zermelo-Fraenkel set theory, IZF

* Extensionality

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

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SLIDE 56

Intuitionistic Zermelo-Fraenkel set theory, IZF

* Extensionality

  • Pairing, Union, Infinity

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

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SLIDE 57

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

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SLIDE 58

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

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SLIDE 59

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

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SLIDE 60

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

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SLIDE 61

Constructive Zermelo-Fraenkel set theory, CZF

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

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SLIDE 62

Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality

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

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SLIDE 63

Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality

  • Pairing, Union, Infinity

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

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SLIDE 64

Constructive Zermelo-Fraenkel set theory, CZF

* Extensionality

  • Pairing, Union, Infinity
  • Bounded Separation

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

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SLIDE 65

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

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SLIDE 66

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

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SLIDE 67

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

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SLIDE 68

Two types of set existence axioms

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

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SLIDE 69

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

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SLIDE 70

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

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SLIDE 71

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

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SLIDE 72

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

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SLIDE 73

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

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SLIDE 74

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

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SLIDE 75

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

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SLIDE 76

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

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SLIDE 77

The Main Theorem

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

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SLIDE 78

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

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SLIDE 79

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

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SLIDE 80

Remarks

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

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SLIDE 81

Remarks

  • This notion of realizability is very robust.

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

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SLIDE 82

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

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SLIDE 83

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

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SLIDE 84

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

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SLIDE 85

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

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SLIDE 86

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

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SLIDE 87

Failure of EP for IZF

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SLIDE 88

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

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SLIDE 89

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

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SLIDE 90

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

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SLIDE 91

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

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SLIDE 92

Problems

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

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SLIDE 93

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

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SLIDE 94

The Weak Existence Property

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

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SLIDE 95

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

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SLIDE 96

Extended E-recursive functions

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

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SLIDE 97

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

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SLIDE 98

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

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SLIDE 99

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

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SLIDE 100

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

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SLIDE 101

Realizability with sets of witnesses

We use the expression a = ∅ 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

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SLIDE 102

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

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

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SLIDE 103

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

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

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SLIDE 104

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

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SLIDE 105

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

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SLIDE 106

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

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SLIDE 107

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

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SLIDE 108

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

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SLIDE 109

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 wEP′.

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

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SLIDE 110

Even better

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

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SLIDE 111

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

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SLIDE 112

Even better

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

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SLIDE 113

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

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SLIDE 114

Conservativity

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SLIDE 115

Conservativity

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

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

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SLIDE 116

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

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SLIDE 117

Theorems and a Conjecture

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

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SLIDE 118

Theorems and a Conjecture

Theorem 1: CZF has the existence property.

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

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SLIDE 119

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

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SLIDE 120

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