Concept

Power set

The collection of all sub-collections of a given set, including the empty one and the whole. It is always strictly larger than the set itself, by the diagonal argument, which is why the sizes of infinite sets never stop increasing.

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

The diagonal, and the row built to be off the list. A table of rows of ones and zeros with the diagonal marked, and beneath it the row obtained by flipping every diagonal entry.

The row that is not on the list

Write down a list of infinite sequences, any list at all, and there is a rule that builds a sequence missing from it. The rule reads one entry from each row, and it is the single most reused argument in this field.

logic · diagonalisation

The tower of sizes, and the gap in it. A ladder of infinite sizes, each the number of sub-collections of the one below, with the space between the first two marked as the one no proof decides.

The size that cannot be pinned down

There is no largest infinity, because no collection has as many members as it has sub-collections. What is not settled is whether anything sits between the first two — and that is not an open problem but a proved absence of an answer.

logic · cardinality

Named alongside it

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

Cardinality Aleph Axiomatic set theory Bijection Consistency Continuum hypothesis Countability Diagonal argument Forcing Independence Model Proof by contradiction

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