Concept

Well ordering

An order on a set in which every non-empty subset has a least element. The whole numbers carry one and the real numbers carry none that anybody has described, and the assertion that every set has one is equivalent to the axiom of choice.

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

The Goodstein sequence from 4, with the ordinal beside each term. A table of the Goodstein sequence with each term's hereditary representation and the ordinal obtained by replacing the base with omega.

A sequence that explodes and still stops

Goodstein's sequence starting at 4 climbs past any number you care to name and reaches zero after about ten to the hundred and twenty million steps. The proof that it stops is a second sequence, running alongside it, that goes down.

logic · ordinals

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

Named alongside it

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

Independence Axiom of choice Choice function Goodstein sequence Hereditary base Maximal element Non measurable set Order type Ordinal Termination Transfinite induction Transfinite recursion

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