Continuum Hypothesis: Proposed by Georg Cantor in 1878, the Continuum Hypothesis (CH) addresses the sizes of infinite sets. It states that there is no set whose cardinality is strictly between that of the integers (countable infinity, ℵ₀) and the real numbers (the continuum, 2^{ℵ₀}). In 1940, Gödel showed CH cannot be disproved from standard set theory (ZFC); in 1963, Cohen proved it cannot be proved either—making CH independent of ZFC. This means CH is neither true nor false within conventional axioms; its truth depends on additional set-theoretic assumptions. The result revolutionized logic and foundations of mathematics, revealing inherent limitations in formal systems. While most mathematicians now view CH as settled in the sense of independence, debates continue about adopting new axioms (e.g., large cardinal axioms) that might resolve it. CH remains a cornerstone of modern set theory and philosophical discussions about mathematical truth.
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