Concept

Axiom of choice

The assumption that one member can be picked from every set in a family at once, with no rule given for the picking. It is independent of the other axioms of set theory, and a surprising number of ordinary theorems — that a countable union of countable sets is countable, among them — quietly need it.

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

Limit ordinals and the sequences that approach them. Several ordinals with the first terms of their fundamental sequences, and the successors marked as having a predecessor instead.

Reached from below, or not at all

Every limit ordinal anybody meets is the end of an increasing sequence — ω, ω·2, ω^ω, all of them approached one step at a time. The first uncountable ordinal is not, and the reason it is not constrains the size of the continuum.

logic · ordinals

Named alongside it

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

Well ordering Cardinality Choice function Cofinality Continuum hypothesis Countability Independence Limit ordinal Maximal element Non measurable set Order type Ordinal

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