Concept

Supremum

The least thing lying above a whole collection, whether or not the collection contains it. A limit ordinal is the supremum of everything below it, and the difference between having a supremum and containing one is what separates a limit from a successor.

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

xⁿ at 5 values of n, and the limit. Several members of the sequence xⁿ drawn on one pair of axes with the function they settle on, and the largest gap between each member and that limit reported.

A limit that forgets to be continuous

Every one of the functions x, x², x³, … is as smooth as anything could be, and every column of the picture settles down. What they settle on has a jump in it — and the quantity that sees the difference is the largest gap anywhere, which is a number about the whole graph rather than about any point of it.

analysis · uniform convergence

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.

Axiom of choice Cardinality Cofinality Continuity Continuum hypothesis Countability Counterexample Function space Gibbs' phenomenon Limit function Limit ordinal Order type

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