**Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?**

Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur GraphSearch.

Concept# Axiome

Résumé

Un axiome (en ἀξίωμα /axioma, « principe servant de base à une démonstration, principe évident en soi » – lui-même dérivé de άξιόω (axioô), « juger convenable, croire juste ») est une proposition non démontrée, utilisée comme fondement d’un raisonnement ou d’une théorie mathématique.
Histoire
Antiquité
Pour Euclide et certains philosophes grecs de l’Antiquité, un axiome était une affirmation qu'ils considéraient comme évidente et qui n'avait nul besoin de démonstration.
Description
Épistémologique
Pour l'épistémologie (branche de la philosophie des sciences), un axiome est une vérité évidente en soi sur laquelle une autre connaissance peut se reposer, autrement dit peut être construite. Tous les épistémologues n'admettent pas que les axiomes, dans ce sens du terme, existent. Dans certains courants philosophiques, comme l'objectivisme, le mot « axiome » a une connotation particulière : un énoncé est axiomatique s'il est impossible de le nie

Source officielle

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées

Chargement

Personnes associées

Chargement

Unités associées

Chargement

Concepts associés

Chargement

Cours associés

Chargement

Séances de cours associées

Chargement

Personnes associées

Publications associées (5)

Aucun résultat

Unités associées

Aucun résultat

Chargement

Chargement

Chargement

Cours associés (16)

CS-550: Formal verification

We introduce formal verification as an approach for developing highly reliable systems. Formal verification finds proofs that computer systems work under all relevant scenarios. We will learn how to use formal verification tools and explain the theory and the practice behind them.

MATH-211: Group Theory

Après une introduction à la théorie des catégories, nous appliquerons la théorie générale au cas particulier des groupes, ce qui nous permettra de bien mettre en perspective des notions telles que quotients de groupe et actions de groupe.

CS-452: Foundations of software

The course introduces the foundations on which programs and programming languages are built. It introduces syntax, types and semantics as building blocks that together define the properties of a program part or a language. Students will learn how to apply these concepts in their reasoning.

Concepts associés (157)

Mathématiques

thumb|upright|Raisonnement mathématique sur un tableau.
Les mathématiques (ou la mathématique) sont un ensemble de connaissances abstraites résultant de raisonnements logiques appliqués à des objets

Logique mathématique

La logique mathématique ou métamathématique est une discipline des mathématiques introduite à la fin du , qui s'est donné comme objet l'étude des mathématiques en tant que langage.
Les objets fond

Théorie des ensembles

La théorie des ensembles est une branche des mathématiques, créée par le mathématicien allemand Georg Cantor à la fin du .
La théorie des ensembles se donne comme primitives les notions d'ensemble e

Séances de cours associées (22)

We are interested in the study of non-correlation of Fourier coefficients of Maass forms against a wide class of real analytic functions. In particular, the class of functions we are interested in should be thought of as some archimedean analogs of Frobenius trace functions. In the first part of the thesis, we give an axiomatic definition for this class, and prove that these functions satisfy properties similar to that of Frobenius trace functions. In particular, we prove non-correlation statements analogous to those given by Fouvry, Kowalski and Michel for algebraic trace functions. In the second part of the thesis, we establish the existence of large values of Hecke-Maass L-functions with prescribed argument. In studying these problems, one encounters sums of Fourier coefficients of Maass forms against real oscillatory functions. In some cases, one can prove that these functions satisfy the axioms discussed previously.

Using arguments developed by De Giorgi in the 1950's, it is possible to prove the regularity of the solutions to a vast class of variational problems in the Euclidean space. The main goal of the present thesis is to extend these results to the more abstract context of metric spaces with a measure. In particular, working in the axiomatic framework of Gol'dshtein – Troyanov, we establish both the interior and the boundary regularity of quasi-minimizers of the p-Dirichlet energy. Our proof works for quite general domains, assuming some natural hypotheses on the (axiomatic) D-structure. Furthermore, we prove analogous results for extremal functions lying in the class of Sobolev functions in the sense of Hajłasz – Koskela, i.e. functions characterized by the single condition that a Poincaré inequality be satisfied. Our strategy to prove these regularity results is first to show that, in a very general setting, the (Hölder) continuity of a function is a consequence of three specific technical hypotheses. This part of the argument is the essence of the De Giorgi method. Then, we verify that for a function u which is a quasi-minimizer in an axiomatic Sobolev space or an extremal Sobolev function in the sense of Hajłasz – Koskela, these technical hypotheses are indeed satisfied and u is thus (Hölder) continuous. In addition to that, we establish the Harnack's inequality for these extremal functions, and we show that the Dirichlet semi-norm of a piecewise-extremal function is equivalent to the sum of the Dirichlet semi-norms of its components.

K-Theory was originally defined by Grothendieck as a contravariant functor from a subcategory of schemes to abelian groups, known today as K0. The same kind of construction was then applied to other fields of mathematics, like spaces and (not necessarily commutative) rings. In all these cases, it consists of some process applied, not directly to the object one wants to study, but to some category related to it: the category of vector bundles over a space, of finitely generated projective modules over a ring, of locally free modules over a scheme, for instance. Later, Quillen extracted axioms that all these categories satisfy and that allow the Grothendieck construction of K0. The categorical structure he discovered is called today a Quillen-exact category. It led him not only to broaden the domain of application of K-theory, but also to define a whole K-theory spectrum associated to such a category. Waldhausen next generalized Quillen's notion of an exact category by introducing categories with weak equivalences and cofibrations, which one nowadays calls Waldhausen categories. K-theory has since been studied as a functor from the category of suitably structured (Quillen-exact, Waldhausen, symmetric monoidal) small categories to some category of spectra1. This has given rise to a huge field of research, so much so that there is a whole journal devoted to the subject. In this thesis, we want to take advantage of these tools to begin studying K-theory from another perspective. Indeed, we have the impression that, in the generalization of topological and algebraic K-theory that has been started by Quillen, something important has been left aside. K-theory was initiated as a (contravariant) functor from the various categories of spaces, rings, schemes, …, not from the category of Waldhausen small categories. Of course, one obtains information about a ring by studying its Quillen-exact category of (finitely generated projective) modules, but still, the final goal is the study of the ring, and, more globally, of the category of rings. Thus, in a general theory, one should describe a way to associate not only a spectrum to a structured category, but also a structured category to an object. Moreover, this process should take the morphisms of these objects into account. This gives rise to two fundamental questions. What kind of mathematical objects should K-theory be applied to? Given such an object, what category "over it" should one consider and how does it vary over morphisms? Considering examples, we have made the following observations. Suppose C is the category that is to be investigated by means of K-theory, like the category of topological spaces or of schemes, for instance. The category associated to an object of C is a sub-category of the category of modules over some monoid in a monoidal category with additional structure (topological, symmetric, abelian, model). The situation is highly "fibred": not only morphisms of C induce (structured) functors between these sub-categories of modules, but the monoidal category in which theses modules take place might vary from one object of C to another. In important cases, the sub-categories of modules considered are full sub-categories of "locally trivial" modules with respect to some (possibly weakened notion of) Grothendieck topology on C . That is, there are some specific modules that are considered sufficiently simple to be called trivial and locally trivial modules are those that are, locally over a covering of the Grothendieck topology, isomorphic to these. In this thesis, we explore, with K-theory in view, a categorical framework that encodes these kind of data. We also study these structures for their own sake, and give examples in other fields. We do not mention in this abstract set-theoretical issues, but they are handled with care in the discussion. Moreover, an appendix is devoted to the subject. After recalling classical facts of Grothendieck fibrations (and their associated indexed categories), we provide new insights into the concept of a bifibration. We prove that there is a 2-equivalence between the 2-category of bifibrations over a category ℬ and a 2-category of pseudo double functors from ℬ into the double category of adjunctions in CAT. We next turn our attention to composable pairs of fibrations , as they happen to be fundamental objects of the theory. We give a characterization of these objects in terms of pseudo-functors ℬop → FIBc into the 2-category of fibrations and Cartesian functors. We next turn to a short survey about Grothendieck (pre-)topologies. We start with the basic notion of covering function, that associate to each object of a category a family of coverings of the object. We study separately the saturation of a covering function with respect to sieves and to refinements. The Grothendieck topology generated by a pretopology is shown to be the result of these two steps. We define then, inspired by Street [89], the notion of (locally) trivial objects in a fibred category P : ℰ → ℬ equipped with some notion of covering of objects of the base ℬ. The trivial objects are objects chosen in some fibres. An object E in the fibre over B ∈ ℬ is locally trivial if there exists a covering {fi : Bi → B}i ∈ I such the inverse image of E along fi is isomorphic to a trivial object. Among examples are torsors, principal bundles, vector bundles, schemes, locally constant sheaves, quasi-coherent and locally free sheaves of modules, finitely generated projective modules over commutative rings, topological manifolds, … We give conditions under which locally trivial objects form a subfibration of P and describe the relationship between locally trivial objects with respect to subordinated covering functions. We then go into the algebraic part of the theory. We give a definition of monoidal fibred categories and show a 2-equivalence with monoidal indexed categories. We develop algebra (monoids and modules) in these two settings. Modules and monoids in a monoidal fibred category ℰ → ℬ happen to form a pair of fibrations . We end this thesis by explaining how to apply this categorical framework to K-theory and by proposing some prospects of research. ______________________________ 1 Works of Lurie, Toën and Vezzosi have shown that K-theory really depends on the (∞, 1)-category associated to a Waldhausen category [94]. Moreover, topological K-theory of spaces and Banach algebras takes the fact that the Waldhausen category is topological in account [62, 70].