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mathematics, in pictures
Concept

Russell paradox

The contradiction produced by the set of all sets that are not members of themselves, which belongs to itself exactly when it does not. It showed that not every describable property forms a set, and it forced set theory onto axioms that say which collections exist.

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

Does this set contain that one — and the row that is missing. A membership table with the diagonal marked, and beneath it the complement of the diagonal, which is not among the rows.

A list that cannot contain itself

The set of all sets that do not contain themselves is not a set. The argument is the diagonal again, applied to a table whose rows and columns are the same objects, and it destroyed the foundations of mathematics in a postcard.

logic · Diagonalisation Adjectives that describe themselves, and the one that cannot exist. A grid of 8 adjectives against the same adjectives as written words, marked where the property holds of the spelling; the diagonal marks self-describing adjectives, and a flipped diagonal row labelled heterological matches no row.

The word that cannot describe itself

Some adjectives describe themselves — 'short' is short — and some do not — 'lengthy' is not lengthy. Call the second kind heterological, and ask whether 'heterological' is heterological. It is exactly when it is not. The diagonal that beat every list of numbers, every set of sets and every provability predicate has been turned on words that describe words, and it proves that no language can contain a word for its own notion of describing, naming or truth.

logic · Diagonalisation

Named alongside it

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

Diagonal argumentSelf-referenceComprehensionConsistencyCountabilityDefinabilityFixed pointFormal systemMembershipObject languageSet theoryTruth

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