epsilon number

Etymology
From the Greek letter, used to denote the numbers.

Noun

 * 1)  Any (necessarily transfinite) ordinal number α such that ωα = α;  any surreal number that is a fixed point of the exponential map x → ωx.
 * 2) * 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.
 * 1) * 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 $$\varepsilon_0$$ (read epsilon nought or epsilon zero), is a limit ordinal definable as the supremum of a sequence of smaller limit ordinals: $$\varepsilon_0 = \omega^{\omega^{\omega^{\cdot^{\cdot^\cdot}}}} = \sup \left\{0, \omega^0=1, \omega^1, \omega^{\omega}, \omega^{\omega^{\omega}}, \omega^{\omega^{\omega^\omega}}, \dots \right\}$$.
 * 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 $$\varepsilon_1, \varepsilon_2,\ldots,\varepsilon_\omega, \varepsilon_{\omega+1}, \ldots, \varepsilon_{\varepsilon_0}, \ldots, \varepsilon_{\varepsilon_1}, \ldots, \varepsilon_{\varepsilon_{\varepsilon_{\cdot_{\cdot_{\cdot}}}}}$$, ...
 * $$\varepsilon_0$$ is countable, as is any $$\varepsilon_\alpha$$ for which $$\alpha$$ 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).

Translations

 * Italian: numero epsilon