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Concept

Axiomatic set theory

The foundation in which every mathematical object is a set and every construction is licensed by a short list of axioms. The usual list is Zermelo and Fraenkel's with choice, and its models can disagree about questions it leaves open, such as the size of the continuum.

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

The tower of sizes, and the gap in it. A tower 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 A countable structure grown by adding witnesses. Stages of a structure built from 0 and 1 by adding sums, products, negatives and roots of quadratics: 2, 4, 12, 158 elements between −3 and 3.

A countable field that passes for the line

The real numbers are uncountable, and every first-order sentence about their addition, multiplication and order is also true of a countable field inside them — the real algebraic numbers. Löwenheim and Skolem showed this is no quirk of the reals: every theory with an infinite model has a countable one, including set theory, which then contains sets it calls uncountable.

logic · Models

Named alongside it

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

CardinalityModelAlephAlgebraic numberCompletenessConsistencyContinuum hypothesisCountabilityElementary equivalenceForcingIndependencePower set

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