## NORMALIZATION OF IZF WITH REPLACEMENT (2007)

### BibTeX

@MISC{07normalizationof,

author = {},

title = {NORMALIZATION OF IZF WITH REPLACEMENT},

year = {2007}

}

### OpenURL

### Abstract

Abstract. IZF is a well investigated impredicative constructive version of Zermelo-Fraenkel set theory. Using set terms, we axiomatize IZF with Replacement, which we call IZFR, along with its intensional counterpart IZF − R. We define a typed lambda calculus λZ corresponding to proofs in IZF − R according to the Curry-Howard isomorphism principle. Using realizability for IZF − R, we show weak normalization of λZ. We use normalization to prove the disjunction, numerical existence and term existence properties. An inner extensional model is used to show these properties, along with the set existence property, for full, extensional IZFR. 1.

### Citations

357 | Set Theory. An introduction to independence proofs - Kunen - 1980 |

280 | Foundations of Constructive Mathematics - Beeson - 1985 |

52 | 2001] Notes on constructive set theory
- Aczel, Rathjen
(Show Context)
Citation Context ...y. Results up to 1985 are presented in [Bee85, ˆ S85]. Later research was concentrated on weaker subsystems [AR01, Lub02]. A predicative constructive set theory CZF has attracted particular interest. =-=[AR01]-=- describes the set-theoretic apparatus available in CZF and provides further references. We axiomatize IZFR, along with its intensional version IZF − R , using set terms. We define a typed lambda calc... |

33 | Typed lambda-calculus in classical Zermelo-Fraenkel set theory, Archive for
- Krivine
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Citation Context ...ut the ∈- induction axiom. This result was later extended — Dowek and Miquel [DM06] interpreted a version of constructive Zermelo set theory in a strongly normalizing deduction-modulo system. Krivine =-=[LK01]-=- defined realizability using lambda calculus for classical set theory conservative over ZF. The types for the calculus were defined. However, it seems that the types correspond more to the truth in th... |

27 |
Realizability and recursive mathematics
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(Show Context)
Citation Context ...FR, along with its intensional version IZF − R , using set terms. We define a typed lambda calculus λZ corresponding to proofs in IZF − R . We also define realizability for IZF − R , in the spirit of =-=[McC84]-=-, and use it to show that λZ weakly normalizes. Strong normalization of λZ does not hold; moreover, we show that in non-well-founded IZF even weak normalization fails. With normalization in hand, the ... |

24 |
The consistency of classical set theory relative to a set theory with intuitionistic logic
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(Show Context)
Citation Context ...ema. We will call the version with Collection IZFC and the version with Replacement IZFR. In the literature, IZF usually denotes IZFC. Both theories extended with excluded middle are equivalent to ZF =-=[Fri73]-=-. 1998 ACM Subject Classification: F.4.1. Key words and phrases: Intuitionistic set theory, Curry-Howard isomorphism, normalization, realizability. Partly supported by NSF grants DUE-0333526 and 04301... |

5 | Extracting programs from constructive HOL proofs via IZF set-theoretic semantics
- Constable, Moczydlowski
- 2006
(Show Context)
Citation Context ... proof IZFR ⊢ n ∈ N. Combine the proofs to get IZFR ⊢ ∃m ∈ N. φ(n,m) and apply NEP to get a number m such that IZFR ⊢ φ(n,m). A detailed account of program extraction from IZFR proofs can be found in =-=[CM06]-=-. There are many provers with the program extraction capability. However, they are usually based on variants of type theory, which is a foundational basis very different from set theory. This makes th... |

4 |
A normalization theorem for set theory
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(Show Context)
Citation Context ...ethod to tackle Leibniz axiom is closer to Friedman’s ideas, in [Moc06b]. 9. Related work Several normalization results for impredicative constructive set theories much weaker than IZFR exist. Bailin =-=[Bai88]-=- proved strong normalization of a constructive set theory without the induction and replacement axioms. Miquel interpreted a theory of similar strength in a Pure Type System [Miq04]. In [Miq03] he als... |

3 | A strongly normalising Curry-Howard correspondence for IZF set theory
- Miquel
(Show Context)
Citation Context ...IZFR exist. Bailin [Bai88] proved strong normalization of a constructive set theory without the induction and replacement axioms. Miquel interpreted a theory of similar strength in a Pure Type System =-=[Miq04]-=-. In [Miq03] he also defined a strongly normalizing lambda calculus with types based on Fω.2, capable of interpreting IZFC without the ∈- induction axiom. This result was later extended — Dowek and Mi... |

2 |
Cut elimination for Zermelo’s set theory. 2006. Manuscript, available from the web pages of the authors
- Dowek, Miquel
(Show Context)
Citation Context ...[Miq03] he also defined a strongly normalizing lambda calculus with types based on Fω.2, capable of interpreting IZFC without the ∈- induction axiom. This result was later extended — Dowek and Miquel =-=[DM06]-=- interpreted a version of constructive Zermelo set theory in a strongly normalizing deduction-modulo system. Krivine [LK01] defined realizability using lambda calculus for classical set theory conserv... |

2 |
and Andre ˆ Sĉedrov. Set existence property for intuitionistic theories with countable choice
- Friedman
- 1983
(Show Context)
Citation Context ...hematical properties of theories related to IZFR are well investigated. Myhill [Myh73] showed DP, NEP, SEP and TEP for IZF with Replacement andnon-recursive list of set terms. Friedman and ˆ Sĉedrov =-=[FS83]-=- showed SEP and TEP for an extension of that theory with countable choice axioms. Recently DP and NEP were shown for IZFC extended with various choice principles by Rathjen [Rat06]. However, the techn... |

1 |
Andre ˆ Sĉedrov. The lack of definable witnesses and provably recursive functions in intuitionistic set theories
- Friedman
- 1985
(Show Context)
Citation Context ...bility. Partly supported by NSF grants DUE-0333526 and 0430161. LOGICAL METHODS IN COMPUTER SCIENCE DOI:10.2168/LMCS-4 (2:1) 2008 c○ Wojciech Moczydłowski ○CC Creative CommonsThey are not equivalent =-=[FS85]-=-. While the proof-theoretic power of IZFC is equivalent to that of ZF, the exact power of IZFR is unknown. Arguably IZFC is less constructive, as Collection, similarly to Choice, asserts the existence... |

1 |
Should your specification language be typed? ACMTOPLAS
- Lamport, Paulson
- 1999
(Show Context)
Citation Context ...foundational basis very different from set theory. This makes the process of formalizing program specification more difficult, as an unfamiliar new language and logic have to be learned from scratch. =-=[LP99]-=- strongly argues against using type theory for the specification purposes, instead promoting standard set theory. IZFR provides therefore the best of both worlds. It is a set theory, with familiar lan... |