Concept

Continuum hypothesis

The statement that there is no size between the whole numbers and the real numbers. It can neither be proved nor disproved from the usual axioms of set theory, so both it and its negation may consistently be assumed, and the size of the continuum is left undetermined by them.

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 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

Limit ordinals and the sequences that approach them. Several ordinals with the first terms of their fundamental sequences, and the successors marked as having a predecessor instead.

Reached from below, or not at all

Every limit ordinal anybody meets is the end of an increasing sequence — ω, ω·2, ω^ω, all of them approached one step at a time. The first uncountable ordinal is not, and the reason it is not constrains the size of the continuum.

logic · ordinals

Named alongside it

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

Cardinality Aleph Axiom of choice Axiomatic set theory Cofinality Consistency Countability Forcing Independence Limit ordinal Model Order type

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