epsilon number

plural epsilon numbers

1. mathematicsAny (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.

   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 ε₀; we can easily show, in fact, that ε₀ is the least epsilon number.

   2 more examples

   More generally, the epsilon numbers are the ordinals #92;varepsilon for which #92;varepsilon#61;#92;omega#92;varepsilon. The smallest epsilon number is #92;varepsilon#95;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.

   1977, Herbert B. Enderton, Elements of Set Theory, Elsevier (Academic Press), page 240

   We show that the associated ordinals are the #92;varepsilon#95;0th epsilon number and the first #92;varepsilon#95;0-critical number, respectively.

   1981, The Journal of Symbolic Logic, Association for Symbolic Logic, page 17