kardinaalset

Kardinaalset is a concept in set theory describing a set whose elements are cardinal numbers. Cardinal numbers index the sizes of sets, including finite numbers and infinite cardinals such as alephs. A kardinaalset is a mathematical device used to discuss collections of sizes and their relationships. In many treatments, a kardinaalset is defined as a nonempty set of cardinals with specified closure properties that depend on the context. Common choices include downward closure (if κ is in the set and λ ≤ κ then λ is in the set) and closure under taking maxima (if κ1 and κ2 are in the set, then max(κ1, κ2) is in the set). Some authors adopt stronger requirements, such as closure under arbitrary directed suprema or under exponentiation by smaller cardinals.

Examples include: the finite kardinaalset {0, 1, 2, ..., n}; the initial segment {κ : κ is a cardinal and κ ≤ κ0} for a fixed κ0; and the set of all infinite cardinals below a given bound within a model.

In use, kardinaalsetes help organize comparisons of sizes of sets, facilitate statements about cardinal arithmetic (addition, multiplication, exponentiation) and support analysis of cofinalities in ordinal constructions. The exact definition and properties of a kardinaalset can vary between texts, and the term is not universally standardized; in many contexts it is simply described as a set of cardinals or as an initial segment of the cardinal order.

See also: cardinal number, aleph numbers, cofinality, set theory, cardinal arithmetic.