In 1874, the German mathematician Georg Cantor asked a question that nobody had thought worth asking: are there more real numbers than rational numbers? Both are infinite, so the intuitive answer is they're the same — both infinite. Cantor showed, decisively, that they are not. There are more reals. Some infinities are bigger than others. The proof — the diagonal argument — fits on a postcard, but it took the mathematical community a generation to forgive him for it. Henri Poincaré called Cantor's work a perverse pathology. The next generation of mathematicians, Hilbert most prominently, called it paradise — and Hilbert promised that no one shall expel us from the paradise that Cantor has created.
Two sets have the same cardinality if and only if there is a bijection between them — a function that pairs every element of one with exactly one element of the other. This is the right definition because it works for finite and infinite sets: a set of three elements has cardinality 3 because it bijects with {1, 2, 3}, and the natural numbers have cardinality ℵ₀ (aleph-null) by definition. Countable sets are those in bijection with ℕ; uncountable sets are those that are infinite but not countable. Cantor showed that the rationals — pairs of integers, seemingly more numerous than the integers — are countable: arrange them in a 2D grid and snake through diagonally. The algebraic numbers (roots of integer polynomials) are countable, by the same kind of construction. The reals, by contrast, are uncountable. Cantor's diagonal argument: assume, for contradiction, that the reals between 0 and 1 are countable; list them in some order with their decimal expansions; construct a new real whose n-th decimal differs from the n-th digit of the n-th listed real; this new real cannot be on the list, contradiction. The cardinality of the reals is 2^ℵ₀, the cardinality of the continuum. Cantor's theorem generalizes: for any set X, the power set P(X) has strictly larger cardinality than X — there is no largest infinity. The hierarchy ℵ₀ < ℵ₁ < ℵ₂ < … goes up forever. The continuum hypothesis (CH) — Cantor's question of whether there is an infinity strictly between ℵ₀ and 2^ℵ₀ — became Hilbert's first problem of 1900. Gödel (1940) showed CH is consistent with ZFC. Paul Cohen (1963) showed its negation is also consistent. CH is therefore independent of ZFC — undecidable in standard set theory, an unsolved problem promoted to an unsolvable one.
Cantor's diagonal is the most-reused proof technique in mathematics. Turing's proof of the undecidability of the Halting Problem is a diagonal argument applied to programs. Gödel's incompleteness theorem is a diagonal argument applied to logic. The cardinality argument also shows that almost every real number is uncomputable: there are countably many programs (each a finite string of symbols) but uncountably many reals, so most reals are unspeakable — they exist, the theory needs them, but no human or computer can ever describe one fully. Type theory in computer science uses cardinality bounds. The continuum hypothesis remains a topic of philosophical debate among set theorists; new axiom candidates (forcing axioms, large cardinal axioms) take stances on it.