kardinaliarvo

Kardinaliarvo, or cardinal value, is a term most often used in mathematics to denote the cardinality or size of a set. The cardinality of a finite set is the exact number of elements it contains, denoted |A|. For infinite sets, cardinal numbers extend this idea to measure how many elements there are in a larger sense. Two sets are equinumerous if there exists a bijection between them, meaning they have the same cardinality.

In set theory, cardinal numbers form a hierarchy. The set of natural numbers has cardinality denoted ℵ0 (aleph-null), which is the smallest infinite cardinal. The set of real numbers has cardinality 2^ℵ0, often called the continuum. Larger infinities exist, such as ℵ1, and the theory explores how these sizes relate to each other. Cantor’s theorem shows that no set and its power set have the same cardinality, demonstrating that infinite cardinals come in distinct levels.

Cardinal arithmetic differs from ordinary arithmetic when infinities are involved. For infinite cardinals κ and λ, the sum and the product satisfy κ + λ = max(κ, λ) and κ · λ = max(κ, λ) under the usual assumptions, though details can depend on foundational choices. The concept of cardinality is central to many areas of mathematics, including combinatorics, analysis, and model theory.

In some contexts outside pure math, kardinaliarvo can also refer to a numerical value on a cardinal (ratio) measurement scale, where both the order of values and the meaningful size of differences, and often a true zero, are considered suitable for arithmetic operations.