Concept

Epsilon nought

The smallest ordinal that cannot be written out of smaller ones using addition, multiplication and powers of omega. It is the limit of the tower ω, ω raised to ω, and so on, and it measures precisely how much induction ordinary arithmetic is able to carry out.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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

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.

Ordinal Countability Decision procedure Diagonalisation Fast growing hierarchy Hereditary base Independence Iteration Limit ordinal Normal form Notation Order type

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