In mathematics, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from A to B such that for every element y of B, there is exactly one element x of A with f(x) = y.
Georg Cantor, the inventor of set theory, showed in 1874 that there is more than one kind of infinity, specifically that the collection of all natural numbers and the collection of all real numbers, while both infinite, are not equinumerous (see Cantor's first uncountability proof).
are equinumerous (an example where a proper subset of an infinite set is equinumerous to the original set), and that the Cartesian product of even a countably infinite number of copies of the real numbers is equinumerous to a single copy of the real numbers.
If the axiom of choice holds, then the cardinal number of a set may be regarded as the least ordinal number of that cardinality (see initial ordinal).
[1] The statement that any two sets are either equinumerous or one has a smaller cardinality than the other is equivalent to the axiom of choice.
If the axiom of choice holds, then the law of trichotomy holds for cardinal numbers, so that any two sets are either equinumerous, or one has a strictly smaller cardinality than the other.
[1] The law of trichotomy for cardinal numbers also implies the axiom of choice.
[3] The Schröder–Bernstein theorem states that any two sets A and B for which there exist two one-to-one functions f : A → B and g : B → A are equinumerous: if |A| ≤ |B| and |B| ≤ |A|, then |A| = |B|.
Cantor's work was harshly criticized by some of his contemporaries, for example by Leopold Kronecker, who strongly adhered to a finitist[5] philosophy of mathematics and rejected the idea that numbers can form an actual, completed totality (an actual infinity).
However, Cantor's ideas were defended by others, for example by Richard Dedekind, and ultimately were largely accepted, strongly supported by David Hilbert.
(aleph naught), the cardinality of any countably infinite set, and the second beth number
A set that is equinumerous to a proper subset of itself is called Dedekind-infinite.
[1][3] The axiom of countable choice (ACω), a weak variant of the axiom of choice (AC), is needed to show that a set that is not Dedekind-infinite is actually finite.
[1] Equinumerosity is compatible with the basic set operations in a way that allows the definition of cardinal arithmetic.
[1] Specifically, equinumerosity is compatible with disjoint unions: Given four sets A, B, C and D with A and C on the one hand and B and D on the other hand pairwise disjoint and with A ~ B and C ~ D then A ∪ C ~ B ∪ D. This is used to justify the definition of cardinal addition.
Furthermore, equinumerosity is compatible with cartesian products: These properties are used to justify cardinal multiplication.
Then the following statements hold: These properties are used to justify cardinal exponentiation.