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is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set. For example, the set of all positive integers is infinite:
In ZF, a set is infinite if and only if the power set of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself. [4] If the axiom of choice is also true, then infinite sets are precisely the Dedekind-infinite sets. If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.
In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees , extensions of Ramsey's theorem , and Martin's axiom .
Indeed, Dedekind defined an infinite set as one that can be placed into a one-to-one correspondence with a strict subset (that is, having the same size in Cantor's sense); this notion of infinity is called Dedekind infinite. Cantor introduced the cardinal numbers, and showed—according to his bijection-based definition of size—that some ...
Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also. As a consequence, the cardinality of the real numbers , which is the same as that of the power set of the integers , is strictly larger than the cardinality of the integers; see Cardinality of the ...
Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas. In discrete mathematics, countable sets (including finite sets) are the main focus
The existence of infinite, Dedekind-finite sets was studied by Bertrand Russell and Alfred North Whitehead in 1912; these sets were at first called mediate cardinals or Dedekind cardinals. With the general acceptance of the axiom of choice among the mathematical community, these issues relating to infinite and Dedekind-infinite sets have become ...
Infinite sets are so common, that when one considers finite sets, this is generally explicitly stated; for example finite geometry, finite field, etc. Fermat's Last Theorem is a theorem that was stated in terms of elementary arithmetic , which has been proved only more than 350 years later.