Kardinaalarv

Kardinaalarv is a mathematical concept used in set theory. It represents the size of a set, essentially counting how many elements are in it. For finite sets, the cardinal number is simply the number of elements. For example, the set {apple, banana, cherry} has a cardinal number of 3.

The concept becomes more significant when dealing with infinite sets. Different infinite sets can have different cardinal numbers, meaning some infinities are "larger" than others. This idea was pioneered by the mathematician Georg Cantor. He introduced the concept of transfinite cardinal numbers to distinguish between these different sizes of infinity.

Cantor proved that the set of natural numbers (1, 2, 3, ...) has the smallest infinite cardinal number, denoted by aleph-null (ℵ₀). He also showed that the set of real numbers has a larger cardinal number, the cardinality of the continuum, often denoted by c. The Continuum Hypothesis, proposed by Cantor, states that there is no cardinal number strictly between ℵ₀ and c. This hypothesis has been proven to be independent of the standard axioms of set theory, meaning it can neither be proven nor disproven within that system. Cardinal numbers are fundamental to understanding the structure and properties of sets, particularly in advanced mathematics.