ordering theorem
31Controversy over Cantor's theory — In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has found wide acceptance in the mathematics community, it has been criticized in several areas by mathematicians and philosophers. Cantor… …
32List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …
33Zorn's lemma — Zorn s lemma, also known as the Kuratowski Zorn lemma, is a proposition of set theory that states:Every partially ordered set in which every chain (i.e. totally ordered subset) has an upper bound contains at least one maximal element.It is named… …
34Ernst Zermelo — Ernst Friedrich Ferdinand Zermelo (July 27 1871, Berlin, German Empire – May 21 1953, Freiburg im Breisgau, West Germany) was a German mathematician, whose work has major implications for the foundations of mathematics and hence on… …
35Zermelo–Fraenkel set theory — Zermelo–Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.ZFC consists of a single primitive ontological notion, that of… …
36Choice function — A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns to each set S in that collection some element f(S) of S. In other words, f is a choice function for X if and… …
37axiom of choice — Math. the axiom of set theory that given any collection of disjoint sets, a set can be so constructed that it contains one element from each of the given sets. Also called Zermelo s axiom; esp. Brit., multiplicative axiom. * * * ▪ set theory… …
38Simple theorems in the algebra of sets — Elementary discrete mathematics courses sometimes leave students under an erroneous impression that the subject matter of set theory is the algebra of union, intersection, and complementation of sets. Those topics are treated below: they would… …
39Dedekind-infinite set — In mathematics, a set A is Dedekind infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind finite if it is not Dedekind… …
40Even and odd ordinals — In mathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers. They are useful in some transfinite induction proofs. The literature contains a few equivalent definitions of the parity of an… …
