Ordinal
Named by 5 essays across one field — each of them below, with the objects they name alongside it.
A sequence that explodes and still stops
Goodstein's sequence starting at 4 climbs past any number you care to name and reaches zero after about ten to the hundred and twenty million steps. The proof that it stops is a second sequence, running alongside it, that goes down.
One step in front of infinitely many
Put one step before an infinite run of them and nothing has changed; put it after and something has. Ordinal addition records that difference, which is why it is not commutative — and why it keeps information that counting throws away.
Every ordinal in base omega
Every ordinal below a certain point is a descending sum of powers of ω, in exactly one way. That notation makes comparison mechanical, it is what hereditary base notation becomes when the base is replaced, and it stops at the first ordinal it cannot name.
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.
An ordinal as a growth rate
Index a family of functions by the ordinals, each one iterating the last, and the index becomes a measure of how fast a function grows. The point where the index leaves what arithmetic can prove is exactly where the Goodstein sequence became unprovable.
Named alongside it
The objects these essays reach for when they reach for this one.
Order typeWell orderingLimit ordinalCardinalityCountabilityEpsilon noughtHereditary baseIndependenceTerminationAssociativityAxiom of choiceCofinality