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Concept

Limit ordinal

An ordinal with things below it and nothing immediately below: the least thing above an increasing collection rather than one step past a predecessor. Induction along the ordinals needs a case of its own for these, and that case is where arithmetic's own strength runs out.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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 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 fast-growing hierarchy at its first few ordinals. A table of the fast-growing hierarchy: one row per ordinal index, one column per argument, with the cells too large to evaluate marked as such.

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.

logic · ordinals

Named alongside it

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

OrdinalCardinalityOrder typeWell orderingAssociativityAxiom of choiceCofinalityCommutativityContinuum hypothesisCountabilityCountingDiagonalisation

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