Cardinal number

A cardinal number expresses the size of a set, measured by which other sets can be paired with it one-to-one. Finite cardinals count elements; infinite cardinals distinguish sizes of infinite sets.

Set: A collection of distinct objects treated as a single mathematical object. Cardinal numbers assign sizes to sets.
Aleph number: A symbol in set theory denoting an infinite cardinal number. Alephs name infinite cardinalities in the hierarchy of well-orderable sets.
Natural number: A number used for counting, conventionally drawn from the nonnegative integers or the positive integers. The natural numbers provide the standard representative of countable infinity.
Ordinal number: A number describing the position or order type of an element in a well-ordered set. Ordinals encode order, while cardinals encode size without regard to arrangement.
Bijection: A function that pairs every element of one set with exactly one element of another, and vice versa. Sets have the same cardinality precisely when a bijection exists between them.
Countable set: A set that is finite or can be put in bijection with the natural numbers. Countability identifies the smallest infinite cardinality, shared by the integers and rationals.
Rational number: A number expressible as a ratio of two integers with nonzero denominator. Despite lying densely on the number line, the rationals are countable.
Measure (mathematics): A function assigning sizes to sets while satisfying specified rules, often including countable additivity. Measures quantify length or probability, unlike cardinality's one-to-one comparison.
Equivalence relation: A relation that is reflexive, symmetric, and transitive. Having the same cardinality partitions sets into size-equivalence classes.
Continuum hypothesis: The claim that no cardinality lies strictly between the natural numbers and the real numbers. It asks whether the continuum is the next cardinal after countable infinity.

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