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Concept

Order type

What is left of an ordered set when the identity of its members is forgotten and only the arrangement is kept. Two sets share an order type when some relabelling matches them in order, and the order types of well-ordered sets are the ordinals.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

The Goodstein sequence from 4, with the ordinal beside each term. A table of the Goodstein sequence with each term's hereditary representation and the ordinal obtained by replacing the base with omega.

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.

logic · Ordinals Ordinal sums and products, in normal form. A table of ordinal expressions with their Cantor normal forms and whether the two sides of each pair are equal, above two tick lines drawing one such pair.

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.

logic · Ordinals The numbers to 18 in hereditary base 2, and their ordinals. A table of small whole numbers written in hereditary base notation beside the ordinal obtained by replacing the base with omega.

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.

logic · Ordinals 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 The order type of a nonstandard model of arithmetic. The ordinary numbers as a run of dots, followed by 9 galaxies — copies of the integers — at the positions of c/2 up to 2c, ordered densely like the rationals, with c² beyond. Infinitely many more galaxies lie between those drawn.

A number larger than every number

Ask for a number bigger than 0, bigger than 1, bigger than 2, and so on for ever. Every finite piece of that request is granted by an ordinary number, so compactness grants all of it at once — in a structure that satisfies every sentence true of the whole numbers and still contains something beyond all of them. Nothing in first-order logic can say 'and nothing else'.

logic · Models

Named alongside it

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

OrdinalWell-orderingCardinalityCountabilityHereditary baseIndependenceLimit ordinalAssociativityAxiomAxiom of choiceCofinalityCommutativity

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