Kardinaliteettiteorian

Kardinaliteettiteoria, or cardinality theory in English, is a branch of mathematics that studies the sizes of sets. It is a fundamental concept in set theory and has applications in various fields of mathematics, including abstract algebra, topology, and logic. The core idea is to define and compare the "number of elements" in sets, even for infinite sets.

Georg Cantor introduced the concept of cardinality in the late 19th century. He demonstrated that not all infinite sets have the same size. This groundbreaking discovery led to the distinction between countable and uncountable infinities. A set is countable if its elements can be put into a one-to-one correspondence with the set of natural numbers. Examples include the set of integers and the set of rational numbers. An uncountable set, on the other hand, cannot be so enumerated. The classic example of an uncountable set is the set of real numbers.

The comparison of cardinalities is done through the concept of bijections. Two sets have the same cardinality if there exists a bijection (a one-to-one and onto function) between them. For infinite sets, Cantor developed a system of transfinite numbers to represent their cardinalities, denoted by Hebrew letters such as aleph (ℵ). The cardinality of the natural numbers is denoted by ℵ₀ (aleph-null), which is the smallest infinite cardinality. The cardinality of the real numbers is denoted by c, or 2 raised to the power of ℵ₀. The Continuum Hypothesis, proposed by Cantor, states that there is no set whose cardinality is strictly between ℵ₀ and c. This hypothesis has been proven to be independent of the standard axioms of set theory (ZFC), meaning it can neither be proven nor disproven within that system.