## Realizability for constructive Zermelo-Fraenkel set theory (2004)

Venue: | STOLTENBERG-HANSEN (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2003 |

Citations: | 5 - 1 self |

### BibTeX

@INPROCEEDINGS{Rathjen04realizabilityfor,

author = {Michael Rathjen},

title = {Realizability for constructive Zermelo-Fraenkel set theory },

booktitle = {STOLTENBERG-HANSEN (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2003},

year = {2004},

pages = {282--314},

publisher = {}

}

### OpenURL

### Abstract

Constructive Zermelo-Fraenkel Set Theory, CZF, has emerged as a standard reference theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. A hallmark of this theory is that it possesses a type-theoretic model. Aczel showed that it has a formulae-as-types interpretation in Martin-Löf’s intuitionist theory of types [14, 15]. This paper, though, is concerned with a rather different interpretation. It is shown that Kleene realizability provides a self-validating semantics for CZF, viz. this notion of realizability can be formalized in CZF and demonstrably in CZF it can be verified that every theorem of CZF is realized. This semantics, then, is put to use in establishing several equiconsistency results. Specifically, augmenting CZF by well-known principles germane to Russian constructivism and Brouwer’s intuitionism turns out to engender theories of equal proof-theoretic strength with the same stock of provably recursive functions.

### Citations

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(Show Context)
Citation Context ... Cantorian mathematics. A hallmark of this theory is that it possesses a type-theoretic model. Aczel showed that it has a formulae-as-types interpretation in Martin-Löf’s intuitionist theory of types =-=[14, 15]-=-. This paper, though, is concerned with a rather different interpretation. It is shown that Kleene realizability provides a self-validating semantics for CZF, viz. this notion of realizability can be ... |

264 |
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Citation Context ...gher order Heyting arithmetic, realizability was first applied to systems of set theory by Myhill [17] and Friedman [11]. More recently, realizability models of set theory were investigated by Beeson =-=[6, 7]-=- (for non-extensional set theories) and McCarty [16] (directly for extensional set theories). [16] is concerned with realizability for intuitionistic Zermelo-Fraenkel set theory, IZF, and employs tran... |

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(Show Context)
Citation Context ..., u) ∧ ∀y ∈ d ∃x ∈ a ψ(x, y, u)]]. The Subset Collection schema easily qualifies as the most intricate axiom of CZF. To explain this axiom in different terms, we introduce the notion of fullness (cf. =-=[1]-=-). Definition: 1.1 As per usual, we use 〈x, y〉 to denote the ordered pair of x and y. We use Fun(g), dom(R), ran(R) to convey that g is a function and to denote the domain and range of any relation R,... |

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Citation Context ...ory algebras, applicative structures, or Schönfinkel algebras. These structures are best described as the models of a theory APP. The following presents the main features of APP; for full details cf. =-=[9, 10, 7, 22]-=-. The language of APP is a first-order language with a ternary relation symbol App, a unary relation symbol N (for a copy of the natural numbers) and equality, =, as primitives. The language has an in... |

82 |
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Citation Context ...ctions. MSC:03F50, 03F35 Keywords: Constructive set theory, realizability, consistency results 1 Introduction Realizability semantics for intuitionistic theories were first proposed by Kleene in 1945 =-=[12]-=-. Inspired by Kreisel’s and Troelstra’s [13] definition of realizability for higher order Heyting arithmetic, realizability was first applied to systems of set theory by Myhill [17] and Friedman [11].... |

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Citation Context ...ance, the case in Bishop’s constructive mathematics (cf.[8]) as well as Brouwer’s intuitionistic analysis (cf.[22], Ch.4, Sect.2). Myhill also incorporated these axioms in his constructive set theory =-=[18]-=-. The weakest constructive choice principle we shall consider is AC ω,ω which asserts that whenever ∀i∈ω ∃j∈ω θ(i, j) then there exists a function f : ω → ω such that ∀i∈ω θ(i, f(i)). The Axiom of Cou... |

46 | Notes on constructive set theory
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(Show Context)
Citation Context ...re is a class J ⊆ ON × V such that I(Φ) = � J α , and for each α, α J α = ΓΦ ( � J β ). J is uniquely determined by the above, and its stages J α will be denoted by Γ α Φ . Proof: [2], section 4.2 or =-=[4]-=-, Theorem 5.1. ✷ β∈α Lemma: 3.4 The classes V(A)α are definable in CZF. Proof: Let Φ be the inductive definition with x a Φ iff ∀u∈a (u ∈ |A| × x). Invoking Lemma 3.3, let J be the class such that I(Φ... |

27 |
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Citation Context ... applied to systems of set theory by Myhill [17] and Friedman [11]. More recently, realizability models of set theory were investigated by Beeson [6, 7] (for non-extensional set theories) and McCarty =-=[16]-=- (directly for extensional set theories). [16] is concerned with realizability for intuitionistic Zermelo-Fraenkel set theory, IZF, and employs transfinite iterations of the powerset operation through... |

16 |
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- 1973
(Show Context)
Citation Context ... [12]. Inspired by Kreisel’s and Troelstra’s [13] definition of realizability for higher order Heyting arithmetic, realizability was first applied to systems of set theory by Myhill [17] and Friedman =-=[11]-=-. More recently, realizability models of set theory were investigated by Beeson [6, 7] (for non-extensional set theories) and McCarty [16] (directly for extensional set theories). [16] is concerned wi... |

14 | The strength of some Martin{Lof type theories. Archive for Mathematical Logic 33
- Rathjen
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(Show Context)
Citation Context ... noted about CZF is: Proposition: 1.3 CZF + EM = ZF. Proof: Note that classically Collection implies Separation. Powerset follows classically from Exponentiation. ✷ On the other hand, it was shown in =-=[20]-=-, Theorem 4.14, that CZF has only the strength of Kripke-Platek Set Theory (with the Infinity Axiom), KP (see [5]), and, moreover, that CZF is of the same strength as its subtheory CZF − , i.e., CZF m... |

11 |
Feferman: Constructive Theories of Functions and classes
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(Show Context)
Citation Context ...ory algebras, applicative structures, or Schönfinkel algebras. These structures are best described as the models of a theory APP. The following presents the main features of APP; for full details cf. =-=[9, 10, 7, 22]-=-. The language of APP is a first-order language with a ternary relation symbol App, a unary relation symbol N (for a copy of the natural numbers) and equality, =, as primitives. The language has an in... |

11 |
Formal systems for some branches of intuitionistic analysis
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(Show Context)
Citation Context ...ive set theory, realizability, consistency results 1 Introduction Realizability semantics for intuitionistic theories were first proposed by Kleene in 1945 [12]. Inspired by Kreisel’s and Troelstra’s =-=[13]-=- definition of realizability for higher order Heyting arithmetic, realizability was first applied to systems of set theory by Myhill [17] and Friedman [11]. More recently, realizability models of set ... |

9 |
An intuitionistic theory of types, predicative
- Martin-Löf
- 1973
(Show Context)
Citation Context ... Cantorian mathematics. A hallmark of this theory is that it possesses a type-theoretic model. Aczel showed that it has a formulae-as-types interpretation in Martin-Löf’s intuitionist theory of types =-=[14, 15]-=-. This paper, though, is concerned with a rather different interpretation. It is shown that Kleene realizability provides a self-validating semantics for CZF, viz. this notion of realizability can be ... |

7 |
Admissible Sets and Structures (Springer-Verlag
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- 1975
(Show Context)
Citation Context ...owerset follows classically from Exponentiation. ✷ On the other hand, it was shown in [20], Theorem 4.14, that CZF has only the strength of Kripke-Platek Set Theory (with the Infinity Axiom), KP (see =-=[5]-=-), and, moreover, that CZF is of the same strength as its subtheory CZF − , i.e., CZF minus Subset Collection. To stay in the world of CZF one has to keep away from any principles that imply EM. Moreo... |

7 | Some properties of Intuitionistic Zermelo-Fraenkel set theory - Myhill - 1973 |

6 |
Continuity in intuitionistic set theories
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- 1979
(Show Context)
Citation Context ...gher order Heyting arithmetic, realizability was first applied to systems of set theory by Myhill [17] and Friedman [11]. More recently, realizability models of set theory were investigated by Beeson =-=[6, 7]-=- (for non-extensional set theories) and McCarty [16] (directly for extensional set theories). [16] is concerned with realizability for intuitionistic Zermelo-Fraenkel set theory, IZF, and employs tran... |

6 | The anti-foundation axiom in constructive set theories
- Rathjen
(Show Context)
Citation Context ...use induction over ω to check that, for all n ∈ ω, This completes the proof of (i). ((n, ρ(n)), ir) � 〈n, (F (n)) s 1〉Kl ∈ F and {e}(ρ(n)) � ϕ((F (n)) s 1, (F (n + 1)) s 1). (ii): RDC implies DC (see =-=[21]-=-, Lemma 3.4) and, on the basis of CZF + DC, RDC follows from the following scheme: ∀x (ϕ(x) → ∃y [ϕ(y) ∧ ψ(x, y)]) ∧ ϕ(b) → (39) ∃z (b ∈ z ∧ ∀x ∈ z ∃y ∈ z [ϕ(y) ∧ ψ(x, y)]). Thus, in view of part (i) ... |

4 | Fragments of Kripke-Platek set theory with infinity - Rathjen - 1992 |