kontinuumhypotézis

The continuum hypothesis, often abbreviated as CH, is a statement in set theory concerning the cardinality of infinite sets. It posits that there is no infinite set whose cardinality is strictly between that of the set of natural numbers and the set of real numbers. In simpler terms, it claims that the smallest infinity is the infinity of the natural numbers, and the next smallest infinity is the infinity of the real numbers, with no sizes of infinity in between.

The cardinality of the set of natural numbers is denoted by aleph-null (ℵ₀), while the cardinality of the set of real numbers is denoted by the cardinality of the continuum (c). The continuum hypothesis states that c = ℵ₁. This means that the real numbers can be put into a one-to-one correspondence with the set of all countable ordinal numbers.

The continuum hypothesis was first formulated by Georg Cantor in 1878. It has a peculiar status in mathematics. Kurt Gödel proved in 1938 that CH is consistent with the standard axioms of set theory (Zermelo-Fraenkel set theory with the axiom of choice, ZFC). This means that assuming CH does not lead to any contradictions within ZFC. Later, Paul Cohen proved in 1963 that the negation of CH is also consistent with ZFC. This result, known as independence, means that CH cannot be proven or disproven from the axioms of ZFC. Therefore, CH is independent of ZFC, and its truth or falsity cannot be determined within this foundational system.