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Concept

Undecidable sentence

A statement that a given list of axioms neither proves nor refutes. Its existence is a theorem about the axioms rather than a report of unfinished work, and it is established by exhibiting two structures obeying the axioms that disagree about the statement.

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 sentence that says it has no proof

Number every sentence and every proof, and a formal system can talk about itself. Then the diagonal is available one more time, and what it builds is a sentence that is true exactly when it is unprovable.

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.

ConsistencyAlephArithmetisationAxiomatic set theoryCardinalityContinuum hypothesisDiagonal argumentFixed pointForcingFormal systemIncompletenessIndependence

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