## A constructive topological proof of van der Waerden's theorem (1993)

Venue: | Journal of Pure and Applied Algebra |

Citations: | 3 - 0 self |

### BibTeX

@ARTICLE{Coquand93aconstructive,

author = {Thierry Coquand},

title = {A constructive topological proof of van der Waerden's theorem},

journal = {Journal of Pure and Applied Algebra},

year = {1993},

volume = {105},

pages = {25125--9}

}

### OpenURL

### Abstract

this paper was written, we became aware of the work [2, 15], which, in the quite different field of functional analysis, illustrates this common idea that localic methods can be used to find sharper reformulations of basic non-constructive results, which become then constructively valid. Our work suggests that these methods, beside the purely mathematical advantage of solving problems concerned with equivariance or continuity in parameters [15], may be interesting also proof-theoretically in providing an elegant framework for extracting computational informations from given mathematical arguments. Our treatment of the topological proof of van der Waerden's theorem is apparently different from the one presented in Girard's book on proof theory [10]. We have not tried though to compare in detail the two arguments, because the main emphasis is somewhat different. The main points of our paper are the formulation of a point-free version of a minimal property, whose ordinary version is proved via Zorn's lemma, and the observation that this point-free version has a direct inductive proof

### Citations

185 |
An introduction to inductive definitions
- Aczel
- 1977
(Show Context)
Citation Context ...pological closure of the set fT n (ff) j n 2 Zg: It is clear that fi is a subcolouring of ff if, and only if, fi belongs to �� ff if, and only if, �� fi is a subset of �� ff. 1 See the sur=-=vey article [1]-=- for a presentation of the theory of inductive definitions. 1.2 The Minimal Property This proposition follows from a standard, non-constructive, use of Zorn's lemma. Proposition (Minimal Property): Fo... |

155 | Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions - Furstenberg - 1977 |

142 |
Elements of Intuitionism
- Dummett
- 1977
(Show Context)
Citation Context ...y infinite sequences n1 ; n2 ; n3 ; : : : there exists p such that n1 : : : np 2 V; but intuitionistically, this equivalence is the content of Brouwer's bar theorem, discussed for instance in Dummett =-=[7]-=- and in our paper [4]. We shall not present a formal proof in the framework of inductive definitions, but instead try to emphasize the basic ideas behind this derivation 4 . Proposition: For any l; th... |

91 | The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis - Gödel - 1938 |

62 |
Intuitionistic formal spaces - a first communication
- Sambin
- 1987
(Show Context)
Citation Context ...nimal bad sequence argument [5]. It should be interesting to develop in a point-free way other notions used in the theory of dynamical systems, maybe following the framework sketched in the reference =-=[16]-=-, and to formulate, for instance, generally in this framework the existence of minimal dynamical systems. Finally, a point-free version of measure and ergodic theory should be also of some interest, a... |

29 |
Topological dynamics and combinatorial number theory
- Furstenberg, Weiss
- 1978
(Show Context)
Citation Context ... here a further illustration of this method by presenting a constructive version of the proof of van der Waerden's theorem on arithmetical progression which uses some elements of topological dynamics =-=[8]-=-. This last non-constructive proof is based on a simple topological lemma, which is a direct corollary of Zorn's lemma (namely, the existence of minimal dynamical system). We formulate a point-free co... |

20 |
A globalization of the Hahn-Banach theorem
- Mulvey, Pelletier
- 1991
(Show Context)
Citation Context ... Contrary to the classical argument however, there is no need to appeal to non-constructive principles in this derivation. After a first version of this paper was written, we became aware of the work =-=[2, 15]-=-, which, in the quite different field of functional analysis, illustrates this common idea that localic methods can be used to find sharper reformulations of basic non-constructive results, which beco... |

16 |
An intuitionistic proof of Tychonoff’s theorem, The
- Coquand
- 1992
(Show Context)
Citation Context ...which is a direct corollary of Zorn's lemma (namely, the existence of minimal dynamical system). We formulate a point-free constructive version of this lemma. As could be expected from other examples =-=[5, 6]-=-, this localic version has a direct (constructive) proof. This may be seen as the first step, in a simple concrete case, of a point-free theory of dynamical systems. From this result, the finite versi... |

13 | Constructive topology and combinatorics
- Coquand
- 1992
(Show Context)
Citation Context ...rmations from given concrete examples of a complex proof of an arithmetical statement. Stimulated by results of de Bruijn, van der Meiden [3], of Martin-Lof [14], and of Johnstone [13], we started in =-=[5]-=- a program of constructivisation of some infinitary proofs in combinatorics (Ramsey's theorem, Higman's lemma) by applying localic methodology. Although the extent to which this yields new results is ... |

8 |
An analysis of Ramsey’s Theorem
- Coquand
- 1994
(Show Context)
Citation Context ...n1 ; n2 ; n3 ; : : : there exists p such that n1 : : : np 2 V; but intuitionistically, this equivalence is the content of Brouwer's bar theorem, discussed for instance in Dummett [7] and in our paper =-=[4]-=-. We shall not present a formal proof in the framework of inductive definitions, but instead try to emphasize the basic ideas behind this derivation 4 . Proposition: For any l; the monotone set of blo... |

3 |
Stone-Cech Compactification of Locales II
- Banaschewski, Mulvey
- 1984
(Show Context)
Citation Context ... Contrary to the classical argument however, there is no need to appeal to non-constructive principles in this derivation. After a first version of this paper was written, we became aware of the work =-=[2, 15]-=-, which, in the quite different field of functional analysis, illustrates this common idea that localic methods can be used to find sharper reformulations of basic non-constructive results, which beco... |