Continuum Hypothesis

The Continuum Hypothesis is the conjecture, formulated by Georg Cantor in 1878, that there is no set with cardinality strictly between that of the natural numbers and that of the real numbers — that the reals are the very next infinite size after the naturals. If ℵ₀ is the cardinality of the naturals, the Continuum Hypothesis asserts that the cardinality of the reals equals ℵ₁, the next cardinal in the hierarchy.

The hypothesis is remarkable for what was proved about it: it is independent of the standard axioms of ZFC set theory. Gödel showed in 1940 that the hypothesis is consistent with ZFC (you cannot disprove it from ZFC). Paul Cohen showed in 1963 that its negation is also consistent with ZFC (you cannot prove it from ZFC). The Continuum Hypothesis is therefore not a question that ZFC can settle. It is, in a precise sense, a question about which mathematical universe we are in — and our axioms do not specify the universe uniquely. Whether this means the hypothesis has no definite truth value, or merely that we have chosen the wrong axioms, is the central dispute in the philosophy of mathematics.

The Continuum Hypothesis was the first of Hilbert's 23 problems in 1900. Its resolution was not the settlement Hilbert imagined but a proof of its unsettlability — a demonstration that mathematical truth outruns mathematical provability in ways Hilbert's formalist program could not absorb.