The Continuum Hypothesis

Set Theory · Axiom Academy

LESSON The Continuum Hypothesis One of mathematics' most famous independent statements We know that ℵ₀ = |ℕ| (the cardinality of the natural numbers) and 𝔠 = 2^ℵ₀ = |ℝ| (the cardinality of the continuum). Cantor proved that ℵ₀ < 𝔠, but asked: In other words: is there a set A such that |ℕ| < |A| < |ℝ|? The Continuum Hypothesis claims no such set exists, meaning the continuum is the "next size up" from countable infinity. The Continuum Hypothesis can be stated in several equivalent ways. Most directly: This says that the cardinality of the continuum (the real numbers) equals ℵ₁, the first uncountable cardinal. Equivalently, there is no cardinal strictly between ℵ₀ and 2^ℵ₀. 3. Generalized Continuum Hypothesis The Generalized Continuum Hypothesis (GCH) extends this pattern to all infinite cardinals. For every ordinal α: This states that for any infinite cardinal, its power set has cardinality equal to the next aleph number. GCH implies CH (by taking α = 0), but is a much stronger statement about the entire hierarchy of infinities. The resolution of CH's status came in two parts: 1940 - Kurt Gödel: Proved that if ZFC is consistent, then ZFC + CH is also consistent. CH cannot be disproved from ZFC. 1963 - Paul Cohen: Proved that if ZFC is consistent, then ZFC + ¬CH is also consistent. CH cannot be proved from ZFC. Together, these results show that CH is independent of ZFC: it is neither provable nor disprovable from the standard axioms of set theory.

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