contiuumhypoteesiin

The Continuum Hypothesis, often referred to in Finnish as "kontiinin hypoteesi" or sometimes mistakenly written as "contiuumhypoteesiin," is a statement in set theory concerning the possible sizes of infinite sets. It asks whether there exists an infinite set whose cardinality is strictly between that of the set of natural numbers and the set of real numbers. In terms of cardinal numbers, the hypothesis can be expressed as: there is no cardinal number \( \kappa \) such that \( \aleph_0 < \kappa < 2^{\aleph_0} \).

The hypothesis was first formulated by Georg Cantor in 1874 as part of his work on the theory of transfinite numbers. Cantor showed that the real numbers form an uncountable set and that the continuum—the cardinality of the real numbers, denoted \( \mathfrak{c} \) or \( 2^{\aleph_0} \)—is strictly larger than the countable infinity \( \aleph_0 \). He asked whether there could be intermediate sizes of infinity, a question that became central to the development of modern set theory.

In 1940, Kurt Gödel proved that the Continuum Hypothesis cannot be disproved from the standard axioms of set theory (Zermelo–Fraenkel set theory with the Axiom of Choice, abbreviated ZFC). Later, in 1963, Paul Cohen showed that it cannot be proved from those axioms either, establishing that the hypothesis is independent of ZFC. Thus, both the assertion that the continuum hypothesis is true and the assertion that it is false are consistent with the usual set‑theoretic foundations, assuming those foundations themselves are consistent.

The independence result has had profound implications for mathematics, illustrating that the axioms of set theory do not determine every mathematical truth. Researchers have explored various extensions of ZFC, such as large cardinal axioms, to examine whether they might settle the Continuum Hypothesis. Some models of set theory honour the hypothesis, while others reject it. The question remains a central topic in mathematical logic and the philosophy of mathematics, highlighting the limits of formal systems in capturing the full structure of the infinite.