Basic concepts in infinite set theory

thumb|318x318px|A one-to-one correspondence between a set of apples and a set of oranges shows they have the same cardinality. In mathematics, cardinality is an inherent property of sets, roughly meaning the number of individual objects they contain, which may be infinite. The concept is understood through one-to-one correspondences between sets. That is, if their objects can be paired such that each object has a pair, and no object is paired more than once.

set with the same cardinality as some subset of the set of natural numbers

hypothesis that no set has a cardinality between that of the integers and that of the real numbers

set with infinite cardinality

set with cardinal number larger than that of the set of all natural numbers

number larger than all finite numbers

In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but is countable, then one says the set is cocountable.

proper subset B of A that is equinumerous to A

constructions of the whole numbers from sets