## Topological Completeness for Higher-Order Logic (1997)

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### Other Repositories/Bibliography

Venue: | Journal of Symbolic Logic |

Citations: | 8 - 3 self |

### BibTeX

@ARTICLE{Awodey97topologicalcompleteness,

author = {Steve Awodey and Carsten Butz},

title = {Topological Completeness for Higher-Order Logic},

journal = {Journal of Symbolic Logic},

year = {1997},

volume = {65},

pages = {65--3}

}

### OpenURL

### Abstract

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces---so-called "topological semantics". The first is classical higherorder logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

### Citations

847 |
A formulation of the simple theory of types
- Church
- 1940
(Show Context)
Citation Context ...heories in classical higher-order logic The systems of classical higher-order logic that we consider are essentially the same as those presented in [4, 11], which in turn are modern formulations 4 of =-=[7]-=-. We summarize one particular formulation for the reader's convenience and for the special purposes of x4. Type symbols are built up inductively from a given list of basic type symbolssX 1 ; : : : ; X... |

188 |
Sheaves in Geometry and Logic. A First Introduction to Topos Theory
- Lane, Moerdijk
- 1992
(Show Context)
Citation Context ... First, one has the universal model UT in the classifying topos BT , as in x2.1. The Grothendieck topos Sh(BT ) of sheaves on BT for the finite epi topology is coherent, and so has enough points (cf. =-=[12]-=-). The covering theorem of the appendix therefore guarantees the existence of a topological space XT and a connected, locally-connected geometric morphism m : Sh(XT ) ! Sh(BT ): The inverse image m : ... |

181 |
Completeness in the Theory of Types
- Henkin
- 1950
(Show Context)
Citation Context ...e are many different spaces which will do the job, depending on various parameters that one is free to choose. We exhibit here just one such choice, intended to be illuminating. To begin, recall from =-=[9]-=- that classical higher-order logic is complete with respect to general models, nowadays called Henkin models. The basic feature of a Henkin model M of a theory T is that a function type Z Y (or power ... |

140 |
P.J.: Introduction to Higher-Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...order logic can be modeled in suitable generalized categories of sets, namely (elementary) topoi, and that with regard to such topos-valued semantics, standard higher-order deduction is complete (see =-=[11]-=- for details). Philosophy Department, Carnegie Mellon University, Pittsburgh PA 15213-3890, USA. awodey@cmu.edu y BRICS, Basic Research in Computer Science, Centre of the Danish National Research Foun... |

30 |
La logique des topos
- Boileau, Joyal
- 1981
(Show Context)
Citation Context ...6, are thanked for supporting our collaboration. 1 Theories in classical higher-order logic The systems of classical higher-order logic that we consider are essentially the same as those presented in =-=[4, 11]-=-, which in turn are modern formulations 4 of [7]. We summarize one particular formulation for the reader's convenience and for the special purposes of x4. Type symbols are built up inductively from a ... |

24 |
Molecular toposes
- Barr, Paré
- 1980
(Show Context)
Citation Context ...erse images p : G ! Sets of these are jointly faithful) , and that a geometric morphism f a fs: E ! F of topoi is connected if the inverse image functor f is full and faithful, and locally connected (=-=[3]: "mo-=-lecular") if f commutes with \Pi-functors. Theorem A. Let Tbe a higher-order theory. There exists a topological space XT and a c-model M of T in Sh(XT ) such that: (i) for any sentence oe 2 L(T),... |

11 |
Logical and cohomological aspects of the space of points of a topos
- Butz
- 1996
(Show Context)
Citation Context ...j'(y; z)g M (as subsheaves of YM \Theta ZM ). Appendix: The spatial cover In the proofs of theorems A and B, use was made of the following covering theorem for topoi, which is part of theorem 13.5 of =-=[5]-=- (also see [6]; cf. [10] for a related result). Covering theorem. For any Grothendieck topos G with enough points there is a topological space X G and a connected, locally connected geometric morphism... |

9 | Topological representation of sheaf cohomology of sites, Compositio Mathematica 118
- Butz, Moerdijk
- 1999
(Show Context)
Citation Context ...s subsheaves of YM \Theta ZM ). Appendix: The spatial cover In the proofs of theorems A and B, use was made of the following covering theorem for topoi, which is part of theorem 13.5 of [5] (also see =-=[6]-=-; cf. [10] for a related result). Covering theorem. For any Grothendieck topos G with enough points there is a topological space X G and a connected, locally connected geometric morphism OE : Sh(X G )... |

4 |
Topological representation of the -calculus
- Awodey
- 1999
(Show Context)
Citation Context ... nature of the models at issue here, logical definability then coincides with continuity in that "minimal" model. For further discussion of this property (in the context of the -calculus) we=-= refer to [1]-=-. Acknowledgments. We have both benefitted greatly from conversations with Ieke Moerdijk on the spatial covering theorem and its logical applications. The Stefan Banach Mathematical Research Center in... |

4 |
Sheaves and logic, Applications of Sheaves
- Fourman, Scott
(Show Context)
Citation Context ...given here is made possible by considering models in topoi other than Sets. The situation is analogous to that of the familiar Heyting-valued completeness theorem for first-order intuitionistic logic =-=[8], which fo-=-llows directly from our theorem B and indeed is the inspiration thereof. Second, and more substantially, the additional "Moreover : : : " clause of each theorem states a further property of ... |

4 |
Topological representation of the λ-calculus
- Awodey
(Show Context)
Citation Context ...nature of the models at issue here, logical definability then coincides with continuity in that “minimal” model. For further discussion of this property (in the context of the λ-calculus) we refer to =-=[1]-=-. Acknowledgments. We have both benefitted greatly from conversations with Ieke Moerdijk on the spatial covering theorem and its logical applications. The Stefan Banach Mathematical Research Center in... |

3 |
Toposes as homotopy groupoids
- Joyal, Moerdijk
- 1990
(Show Context)
Citation Context ...ves of YM \Theta ZM ). Appendix: The spatial cover In the proofs of theorems A and B, use was made of the following covering theorem for topoi, which is part of theorem 13.5 of [5] (also see [6]; cf. =-=[10]-=- for a related result). Covering theorem. For any Grothendieck topos G with enough points there is a topological space X G and a connected, locally connected geometric morphism OE : Sh(X G ) ! G: Thus... |