epsilon number

English

Etymology

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

Noun

epsilon number (plural epsilon numbers)

  1. (set theory) Any (necessarily transfinite) ordinal number α such that ωα = α; (by generalisation) any surreal number that is a fixed point of the exponential map x → ωx.
    • 1977, Herbert B. Enderton, Elements of Set Theory, Elsevier (Academic Press), page 240:
      More generally, the epsilon numbers are the ordinals for which . The smallest epsilon number is . 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 th epsilon number and the first -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 ; we can easily show, in fact, that is the least epsilon number.

Usage notes

  • The smallest epsilon number, denoted (read epsilon nought or epsilon zero), is a limit ordinal definable as the supremum of a sequence of smaller limit ordinals: .
    • This sequence can be extended recursively: , , , ...
    • The recursion is applied transfinitely, thus extending the definition to , ...
  • is countable, as is any for which 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

Translations

See also

  • epsilon-induction

Further reading

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