epsilon number

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Epsilon numbers (mathematics)

Wikipedia

Etymology

From the Greek letter ε (“epsilon”), used to denote the numbers.

Noun

epsilon number (plural epsilon numbers)

(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 → ω^x.
- 1977, Herbert B. Enderton, Elements of Set Theory‎^[1], Elsevier (Academic Press), page 240:
  More generally, the epsilon numbers are the ordinals $\varepsilon$ for which $\varepsilon =\omega ^{\varepsilon }$ . The smallest epsilon number is $\varepsilon _{0}$ . 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.
- 1981, The Journal of Symbolic Logic, Association for Symbolic Logic, page 17:
  We show that the associated ordinals are the $\varepsilon _{0}$ th epsilon number and the first $\varepsilon _{0}$ -critical number, respectively.
- 2014, Charles C. Pinter, A Book of Set Theory, 2014, Dover, [Revision of 1971 Addison-Wesley edition], page 203,
  Thus there is at least one epsilon number, namely $\varepsilon _{0}$ ; we can easily show, in fact, that $\varepsilon _{0}$ is the least epsilon number.

Usage notes

The smallest epsilon number, denoted $\varepsilon _{0}$ $\varepsilon _{0}$ (read epsilon nought or epsilon zero), is a limit ordinal definable as the supremum of a sequence of smaller limit ordinals: $\varepsilon _{0}=\omega ^{\omega ^{\omega ^{\cdot ^{\cdot ^{\cdot }}}}}=\sup \left\{0,\omega ^{0}=1,\omega ^{1},\omega ^{\omega },\omega ^{\omega ^{\omega }},\omega ^{\omega ^{\omega ^{\omega }}},\dots \right\}$ $\varepsilon _{0}=\omega ^{\omega ^{\omega ^{\cdot ^{\cdot ^{\cdot }}}}}=\sup \left\{0,\omega ^{0}=1,\omega ^{1},\omega ^{\omega },\omega ^{\omega ^{\omega }},\omega ^{\omega ^{\omega ^{\omega }}},\dots \right\}$ .
- This sequence can be extended recursively: $\varepsilon _{1}=\sup \left\{{\varepsilon _{0}+1},\omega ^{\varepsilon _{0}+1},\omega ^{\omega ^{\varepsilon _{0}+1}},\dots \right\}$ , $\varepsilon _{2}=\sup \left\{{\varepsilon _{1}+1},\omega ^{\varepsilon _{1}+1},\omega ^{\omega ^{\varepsilon _{1}+1}},\dots \right\}$ , $\varepsilon _{3}=\sup \left\{{\varepsilon _{2}+1},\omega ^{\varepsilon _{2}+1},\omega ^{\omega ^{\varepsilon _{2}+1}},\dots \right\}$ , ...
- The recursion is applied transfinitely, thus extending the definition to $\varepsilon _{1},\varepsilon _{2},\ldots ,\varepsilon _{\omega },\varepsilon _{\omega +1},\ldots ,\varepsilon _{\varepsilon _{0}},\ldots ,\varepsilon _{\varepsilon _{1}},\ldots ,\varepsilon _{\varepsilon _{\varepsilon _{\cdot _{\cdot _{\cdot }}}}}$ , ...
$\varepsilon _{0}$ $\varepsilon _{0}$ is countable, as is any $\varepsilon _{\alpha }$ $\varepsilon _{\alpha }$ for which $\alpha$ $\alpha$ is countable.
- Epsilon numbers also exist that are uncountable; the index of any such must itself be an uncountable ordinal.
When generalised to the surreal number domain, epsilon numbers are no longer required to be ordinals and the index may be any surreal number (including any negative, fraction or limit).