In mathematics, a set is countable if it has the same cardinality (the number of elements of the set) as some subset of the set of natural numbers N = {0, 1, 2, 3, ...}.

A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton.

In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. This inductive definition is well-founded and can be expressed in the language of first-order set theory.

A box set or boxed set is a set of items (for example, a compilation of books, musical recordings, films or television programs) traditionally packaged in a box and offered for sale as a single unit.

In tennis, a golden set is a set which is won without losing a single point. This means scoring the 24 minimum points required to win the set 6–0, without conceding any points.

In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in German with G for [(German: area, or neighbourhood) meaning open set in this case and for Durchschnitt] (German: intersection)..

The term set piece or set play is used in association football and rugby football to refer to a situation when the ball is returned to open play, for example following a stoppage, particularly in a forward area of the pitch. In association football, the term usually refers to free kicks and corners, but sometimes penalties and throw-ins.

The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory. It is one of Karp's 21 NP-complete problems shown to be NP-complete in 1972.

<mathematics, graphics> (After its discoverer, {Benoit Mandelbrot}) The set of all complex numbers c such that | z[N] | < 2 for arbitrarily large values of N, where z[0] = 0 z[n+1] = z[n]^2 + c The Mandelbrot set is usually displayed as an {Argand diagram}, giving each point a colour which depends on the largest N for which | z[N] | < 2, up to some maximum N which is used for the points in the set (for which N is infinite). These points are traditionally coloured black. The Mandelbrot set is the best known example of a fractal - it includes smaller versions of itself which can be explored to arbitrary levels of detail. {The Fractal Microscope (http://ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html/)}. (1995-02-08)

<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)