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mathematics, in pictures
Concept

Infinity

The size of a collection that never runs out, such as the whole numbers or the points of a line. Arguments that work step by step for finite collections can fail for infinite ones, and repairing them usually needs a finiteness condition somewhere else.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Infinitely many guessers, finitely many wrong. Three rows over the first 40 places: the hats worn, the chosen representative of their class, and a row of marks showing each guess right or wrong. The 5 wrong guesses all fall within the first 14 places, up to a marked place; every later guess is right.

Infinitely many guessers, finitely many wrong

An infinite line of people each wears a black or white hat, sees every hat in front and none of their own, and must guess their own colour. With a finite line, each guesser is right half the time whatever they agree in advance. With an infinite line and the axiom of choice, they can agree a strategy under which all but finitely many are right — and nobody can carry it out.

logic · Axiom of choice An infinite graph that passes every finite test. Two columns of vertices continuing downward without end. The top left vertex joins every vertex on the right; each other left vertex joins only the one beside it.

Enough partners in every finite group

Hall's theorem says a matching exists unless some group has too few partners between them. On an infinite graph that is false: one vertex that can be paired with anybody, and every other with exactly one, passes the test on every finite group and has no matching at all. The failure needs a vertex with infinitely many choices. Forbid that, and the theorem comes back, by an argument about trees rather than about matchings.

discrete · Halls theorem

Named alongside it

The objects these essays reach for when they reach for this one.

Axiom of choiceBipartite graphChoice functionCompactnessEquivalence classExhaustive searchIndependenceMatchingNon-measurable setParityPigeonhole principleProbability

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