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Concept

Choice function

A function that picks one element out of each set in a given family of non-empty sets. The axiom of choice asserts that one always exists, even when the family is infinite and no rule describes which element to pick.

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

Every way of choosing one thing from each of 4 pairs. A table with one row per choice function on a small family of pairs, each row giving what it takes from each pair, with the row a stated rule names picked out.

The choice nobody can write down

Given finitely many pairs, picking one thing from each is a finite list of decisions and needs no justification. Given infinitely many, the list cannot be finished — and whether one exists anyway is an axiom, independent of everything else, whose consequences include a theorem most people refuse to believe.

logic · Axiom of choice 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

Named alongside it

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

Axiom of choiceIndependenceNon-measurable setEquivalence classExhaustive searchInfinityMaximal elementParityProbabilityTransfinite recursionWell-orderingZorns lemma

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