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Ordinals — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · logic
  2. 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.

    part 2 · logic
  3. 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.

    part 3 · logic
  4. 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.

    part 4 · logic
  5. 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.

    part 5 · logic

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