In
mathematics, axiomatic set theory is a
rigorous reformulation of
set theory in
first-order logic created to address paradoxes in
naive set theory. The basis of set theory was created principally by the
German mathematician Georg Cantor at the end of the 19th century.Initially controversial, set theory has come to play the role of a
foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (numbers, functions, etc.,) from all the traditional areas of mathematics (
algebra,
analysis,
topology, etc.) in a single theory, and provides a standard set of axioms to prove or disprove them. At the same time the basic concepts of set theory are used throughout mathematics, the subject is pursued in its own right as a specialty by a comparatively small group of mathematicians and
logicians. It should be mentioned that there are also mathematicians using and promoting different approaches to the foundations of mathematics.
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<
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 Frnkel 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.
NF ("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)
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