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Concept

Hereditary base

A number written in a base with its exponents written in that base too, and their exponents likewise, until every digit visible is smaller than the base. Replacing the base by a larger one then leaves the shape of the expression unchanged, which is what a Goodstein sequence is built on.

Named by 2 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 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

Named alongside it

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

Order typeOrdinalWell orderingCountabilityDecision procedureEpsilon noughtGoodstein sequenceIndependenceNormal formNotationTerminationTransfinite induction

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