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Concept

Definability

The property of a set or relation that some formula of a given language picks out exactly its members. Whether a set is definable depends on the language, and counting shows most sets of numbers are definable in none, since formulas can be listed and sets cannot.

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

Axioms, the conditions on the arrows they answer to, and the one that answers to none. A table of modal axioms with the property of the accessibility relation each corresponds to, every row decided by sweeping all relations on up to four worlds.

The axiom with no property of the arrows

Each axiom of modal logic can be matched by hand to a condition on the arrows between worlds. There is a recipe that does it for a whole class of axioms, and there is an axiom the recipe cannot reach — not because nobody has looked, but because no condition on the arrows defines it at all.

logic · Modal logic 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.

AccessibilityAxiomCountabilityDiagonal argumentExhaustive searchExpressive powerFixed pointFormal systemFrameKripke modelModal logicObject language

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