## Läuchli’s completeness theorem from a topos-theoretic (2007)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Menni07läuchli’scompleteness,

author = {Matías Menni},

title = {Läuchli’s completeness theorem from a topos-theoretic},

year = {2007}

}

### OpenURL

### Abstract

perspective

### Citations

745 | Introduction to Metamathematics - Kleene - 1952 |

198 |
Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Springer-Verlag
- Lane, Moerdijk
- 1994
(Show Context)
Citation Context ...y work with two types of hyperdoctrines (always over toposes). The first type is the standard hyperdoctrine which assigns to each object X in the underlying topos E, the poset of subobjects of X (see =-=[11, 3]-=-). The second type is the one determined by the assignment X ↦→ E/X. This hyperdoctrine will be denoted by pE. When interpreting formulas in pE, relation symbols of type X are interpreted as maps with... |

107 |
Semantical analysis of intuitionistic logic, I
- Kripke
- 1965
(Show Context)
Citation Context ... states de following. Lemma 2.2. Let A ∈ F (ΨΛ) with not ⊢ A. Then (and only then) there is a model Φ such that Φ(A, Λ) = F . He then comments that the proof of Lemma 2.2 “is clear from Kripke’s work =-=[7]-=-. The element U is no bother: Any Φ which is defined on Σ − {U} can be extended to Σ by setting Φ(A, U) = T for all A ∈ F (ΨU)”. The extent to which the proof is clear from Kripke’s work may be a matt... |

91 | Categorical logic
- Pitts
- 1996
(Show Context)
Citation Context ... Läuchli’s result in some detail let us go back to its categorical formulation. We will assume that the reader is familiar with the interpretation of first order logic in hyperdoctrines (see [10] and =-=[14]-=-). We will only work with two types of hyperdoctrines (always over toposes). The first type is the standard hyperdoctrine which assigns to each object X in the underlying topos E, the poset of subobje... |

56 | Equality in hyperdoctrines and the comprehension schema as an adjoint functor - Lawvere - 1970 |

37 |
Some free constructions in realizability and proof theory
- Carboni
- 1995
(Show Context)
Citation Context ... Z ) ex is complete. General abstract nonsense about exact completions then implies Theorem 3.5. 4.1 The exact completion In this section we recall the facts about exact completions that we need. See =-=[1]-=- and [13] for the details. Proposition 4.1. Let Cex be the exact completion of a category with finite limits C. 1. There is a full embedding y : C → Cex whose image is essentially the subcategory of p... |

23 | Adjointness in foundations
- Lawvere
- 1969
(Show Context)
Citation Context ...hli-counter-models is (essentially) the inverse image of a geometric morphism. Completeness follows because this geometric morphism is an open surjection. 1 Introduction In the author’s commentary of =-=[10]-=- Lawvere mentions “Läuchli’s 1967 success in finding a completeness theorem for Heyting predicate calculus lurking in the category of ordinary permutations” as inspiration for the introduction of hype... |

18 |
An abstract notion of realizability for which intuitionistic predicate calculus is complete
- Lauchli
- 1970
(Show Context)
Citation Context ...category of ordinary permutations” as inspiration for the introduction of hyperdoctrines. In the introduction to [9] he also comments that hyperdoctrines appear to be related to Läuchli’s result (see =-=[8]-=-) but that “the precise relation is yet to be worked out”. The first attempts to establish the precise relation were in lectures by Lawvere “at the AMS Los Angeles meeting in August 1967 as well as an... |

14 | Exact completions and toposes - Menni - 2000 |

6 |
Lambek’s categorical proof theory and Läuchli’s abstract realizability
- Harnik, Makkai
- 1992
(Show Context)
Citation Context ...used to give a uniform treatment of completeness theorems for intuitionistic predicate calculus. In this framework, Läuchli’s result also appears as a corollary of a representation theorem. (See also =-=[2]-=-.) Both in [12] and [5] the structure used to study Läuchli’s work is the hyperdoctrine determined by the canonical indexing of the topos Set Z of Z-sets. That is, the hyperdoctrine determined by the ... |

3 |
The fibrational formulation of intuitionistic prediate logic 1: completeness according to Godel, Kripke and Lauchli. Part 2. Notre Dame journal of formal logic
- Makkai
- 1993
(Show Context)
Citation Context ... participants of the seminar “much-needed encouragement” that the ideas being developed were “potentially fruitful” [6].) The relation between Läuchli’s theorem and hyperdoctrines is also explored in =-=[12]-=- where fibrations are used to give a uniform treatment of completeness theorems for intuitionistic predicate calculus. In this framework, Läuchli’s result also appears as a corollary of a representati... |

1 |
On a theorem of Läuchli concerning proof bundles. Unpublished
- Kock
- 1970
(Show Context)
Citation Context ... Angeles meeting in August 1967 as well as another AMS meeting in New York (which became the ‘hyperdoctrines’ paper) and at the 1968 Batelle meeting in Seattle” [6]. Further work in this direction is =-=[5]-=- which unfortunatly was never published. There, Kock uses pre-doctrines to formulate and prove a variant of Läuchli’s result in the form of a pre-doctrinal representation theorem. (Kock’s work was don... |

1 | Private communication - Kock, Lawvere |