Cardinality
Cardinality is the number of elements in a set, generalized to infinite sets by comparing sets through one-to-one correspondences.
Bijection: A function that pairs every element of its domain with exactly one element of its codomain, and vice versa. A bijection is the criterion for two sets to have the same cardinality.
Set: A collection of distinct objects treated as a single mathematical entity. Cardinality assigns a size to sets, regardless of what their elements are.
Natural numbers: The numbers used for counting, usually beginning with zero or one. Their cardinality is the standard smallest infinite size.
Georg Cantor: A German mathematician who founded set theory and developed the theory of transfinite numbers. His investigations of trigonometric series led to systematic comparisons of infinite sets.
Ordinal number: A number describing the position type of a well-ordered set. Ordinals encode order structure, while cardinalities ignore the arrangement of elements.
Injection: A function that maps distinct elements of its domain to distinct elements of its codomain. An injection from one set into another expresses that the first is no larger.
Function: A relation assigning exactly one output to each input in a specified domain. Functions provide the mappings used to compare the sizes of sets.
Integers: The whole numbers extending indefinitely in both positive and negative directions, including zero. Despite including negatives, they can be listed in a sequence like the natural numbers.
Set theory: The mathematical study of sets, their elements, and relations between them. Cardinality became a central measure within Cantor's developing theory of sets.
Measure theory: The mathematical study of assigning consistent sizes to sets, including lengths, areas, and probabilities. Measure captures geometric size that cardinality cannot distinguish between most subsets of the real line.