Last Updated：2025/12/05

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(set theory) Any (necessarily transfinite) ordinal number α such that ω^α = α; (by generalisation) any surreal number that is a fixed point of the exponential map x → ωˣ.

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epsilon number

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Source Word

epsilon number

Noun

(set theory) Any (necessarily transfinite) ordinal number α such that ω^α = α; (by generalisation) any surreal number that is a fixed point of the exponential map x → ωˣ.

Japanese Meaning

セット理論において、ω^α = α を満たす（必然的に超限の）順序数としてのイプシロン数 / 一般化して、指数写像 x → ωˣ の不動点となる超実数あるいは超順序数としてのイプシロン数

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Google Translation / DeepL Translation

ε数とは ω^α = α を満たす（必ず超限の）順序数 α のことであり、その研究は順序数の累乗に関する深い性質を明らかにする。

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Sense（1）

(set theory) Any (necessarily transfinite) ordinal number α such that ω^α = α; (by generalisation) any surreal number that is a fixed point of the exponential map x → ωˣ.

More generally, the epsilon numbers are the ordinals ε for which ε=𝜔^ε. The smallest epsilon number is ε₀. It is a countable ordinal, being the countable union of countable sets. By the Veblen fixed-point theorem, the class of epsilon numbers is unbounded.

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epsilon numbers

（ plural ）

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epsilon number

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epsilon number

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