epsilon number

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Epsilon numbers (mathematics)

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

epsilon number (plural epsilon numbers)

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 → ω^x.
   • 1977, Herbert B. Enderton, Elements of Set Theory‎, Elsevier (Academic Press), page 240:

     More generally, the epsilon numbers are the ordinals ${\displaystyle \varepsilon }$ for which ${\displaystyle \varepsilon =\omega ^{\varepsilon }}$. The smallest epsilon number is ${\displaystyle \varepsilon _{0}}$. 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 ${\displaystyle \varepsilon _{0}}$th epsilon number and the first ${\displaystyle \varepsilon _{0}}$-critical number, respectively.

   • 2014, Charles C. Pinter, A Book of Set Theory, 2014, Dover, , page 203,

     Thus there is at least one epsilon number, namely ${\displaystyle \varepsilon _{0}}$; we can easily show, in fact, that ${\displaystyle \varepsilon _{0}}$ is the least epsilon number.

• The smallest epsilon number, denoted ${\displaystyle \varepsilon _{0}}$ (read epsilon nought or epsilon zero), is a limit ordinal definable as the supremum of a sequence of smaller limit ordinals: ${\displaystyle \varepsilon _{0}=\omega ^{\omega ^{\omega ^{\cdot ^{\cdot ^{\cdot }}}}}=\sup \left\{0,\omega ^{0}=1,\omega ^{1},\omega ^{\omega },\omega ^{\omega ^{\omega }},\omega ^{\omega ^{\omega ^{\omega }}},\dots \right\}}$.
  • This sequence can be extended recursively: ${\displaystyle \varepsilon _{1}=\sup \left\{{\varepsilon _{0}+1},\omega ^{\varepsilon _{0}+1},\omega ^{\omega ^{\varepsilon _{0}+1}},\dots \right\}}$, ${\displaystyle \varepsilon _{2}=\sup \left\{{\varepsilon _{1}+1},\omega ^{\varepsilon _{1}+1},\omega ^{\omega ^{\varepsilon _{1}+1}},\dots \right\}}$, ${\displaystyle \varepsilon _{3}=\sup \left\{{\varepsilon _{2}+1},\omega ^{\varepsilon _{2}+1},\omega ^{\omega ^{\varepsilon _{2}+1}},\dots \right\}}$, ...
  • The recursion is applied transfinitely, thus extending the definition to ${\displaystyle \varepsilon _{1},\varepsilon _{2},\ldots ,\varepsilon _{\omega },\varepsilon _{\omega +1},\ldots ,\varepsilon _{\varepsilon _{0}},\ldots ,\varepsilon _{\varepsilon _{1}},\ldots ,\varepsilon _{\varepsilon _{\varepsilon _{\cdot _{\cdot _{\cdot }}}}}}$, ...
• ${\displaystyle \varepsilon _{0}}$ is countable, as is any ${\displaystyle \varepsilon _{\alpha }}$ for which ${\displaystyle \alpha }$ 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).

• delta number
• gamma number

• epsilon-induction

• Ordinal arithmetic on Wikipedia.Wikipedia
• Additively indecomposable ordinal on Wikipedia.Wikipedia
• Large countable ordinal on Wikipedia.Wikipedia
• Veblen function on Wikipedia.Wikipedia