Concept

Zorns lemma

The statement that a partially ordered set in which every chain has an upper bound contains a maximal element. It is equivalent to the axiom of choice, and it is the usual way that axiom is used to build bases, maximal ideals and other objects no construction can name.

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

An additive function that is nowhere a line. A square window with 1157 points of the graph of an additive function that sends √2 to 0, scattered across the window and meeting every part of it, beside a dashed diagonal marking the straight line y = x.

A function that adds and is nowhere a line

Every continuous function with f(x + y) = f(x) + f(y) is a straight line through the origin. Drop continuity and, given the axiom of choice, there are others — functions that add perfectly and whose graphs are scattered densely over the whole plane. A finite piece of the construction can be drawn exactly; the whole of it needs a basis of the real numbers that no one can write down.

logic · Axiom of choice

Named alongside it

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

Axiom of choice Independence Non-measurable set Basis Choice function Continuity Dense set Irrationality Linearity Maximal element Transfinite recursion Well-ordering

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