<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. 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)

<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)

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music.

In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.

Set theory is a branch of mathematics concerning mathematical sets. A set theory may also refer to a specific formalization of the mathematics of sets; for a listing of different alternative set theories in this sense, see alternative set theory.

In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG). While von Neumann–Bernays–Gödel set theory restricts the bound variables in the schematic formula appearing in the axiom schema of Class Comprehension to range over sets alone, Morse–Kelley set theory allows these bound variables to range over proper classes as well as sets, as first suggested by Quine in 1940 for his system ML.

<mathematics> Zermelo Frankel set theory plus the {Axiom of Choice}. A favourite axiomatisation of set theory. (1995-03-29)

The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek.

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. and in a review by Laszlo Kalmar ().