axiomatic set theory - définition. Qu'est-ce que axiomatic set theory
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Qu'est-ce (qui) est axiomatic set theory - définition

BRANCH OF MATHEMATICS THAT STUDIES SETS, WHICH ARE COLLECTIONS OF OBJECTS
Axiomatic Set Theory; SetTheory; Set Theory; Formal set theory; Axiomatic set theory; Theory of sets; Ordinary set theory; Set theorist; Classical set theory; Set-theoretic; Axioms of set theory; Axiom of set theory; Axiomatic set theories; Transfinite set theory; Abstract set theory; Mathematical set theory; Set theory (mathematics); Applications of set theory; History of set theory
  • [[Georg Cantor]]

axiomatic set theory         
<theory> One of several approaches to set theory, consisting of a formal language for talking about sets and a collection of axioms describing how they behave. There are many different axiomatisations for set theory. Each takes a slightly different approach to the problem of finding a theory that captures as much as possible of the intuitive idea of what a set is, while avoiding the paradoxes that result from accepting all of it, the most famous being Russell's paradox. The main source of trouble in naive set theory is the idea that you can specify a set by saying whether each object in the universe is in the "set" or not. Accordingly, the most important differences between different axiomatisations of set theory concern the restrictions they place on this idea (known as "comprehension"). Zermelo Frankel set theory, the most commonly used axiomatisation, gets round it by (in effect) saying that you can only use this principle to define subsets of existing sets. NBG (von Neumann-Bernays-Goedel) set theory sort of allows comprehension for all formulae without restriction, but distinguishes between two kinds of set, so that the sets produced by applying comprehension are only second-class sets. NBG is exactly as powerful as ZF, in the sense that any statement that can be formalised in both theories is a theorem of ZF if and only if it is a theorem of ZFC. MK (Morse-Kelley) set theory is a strengthened version of NBG, with a simpler axiom system. It is strictly stronger than NBG, and it is possible that NBG might be consistent but MK inconsistent. axiomatic set theoryholmes/holmes/nf.html">NF (http://math.boisestate.edu/axiomatic set theoryholmes/holmes/nf.html) ("New Foundations"), a theory developed by Willard Van Orman Quine, places a very different restriction on comprehension: it only works when the formula describing the membership condition for your putative set is "stratified", which means that it could be made to make sense if you worked in a system where every set had a level attached to it, so that a level-n set could only be a member of sets of level n+1. (This doesn't mean that there are actually levels attached to sets in NF). NF is very different from ZF; for instance, in NF the universe is a set (which it isn't in ZF, because the whole point of ZF is that it forbids sets that are "too large"), and it can be proved that the Axiom of Choice is false in NF! ML ("Modern Logic") is to NF as NBG is to ZF. (Its name derives from the title of the book in which Quine introduced an early, defective, form of it). It is stronger than ZF (it can prove things that ZF can't), but if NF is consistent then ML is too. (2003-09-21)
set theory         
<mathematics> A mathematical formalisation of the theory of "sets" (aggregates or collections) of objects ("elements" or "members"). Many mathematicians use set theory as the basis for all other mathematics. Mathematicians began to realise toward the end of the 19th century that just doing "the obvious thing" with sets led to embarrassing paradoxes, the most famous being {Russell's Paradox}. As a result, they acknowledged the need for a suitable axiomatisation for talking about sets. Numerous such axiomatisations exist; the most popular among ordinary mathematicians is Zermelo Frankel set theory. {set theoryhistory/HistoryTopics.html">The beginnings of set theory (http://www-groups.dcs.st-and.ac.uk/set theoryhistory/HistoryTopics.html)}. (1995-05-10)
set theory         
¦ noun the branch of mathematics concerned with the formal properties and applications of sets.

Wikipédia

Set theory

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.

The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied.

Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science (such as in the theory of relational algebra), philosophy and formal semantics. Its foundational appeal, together with its paradoxes, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for logicians and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.