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In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ε that satisfy the equation
in which ω is the smallest infinite ordinal.
The least such ordinal is ε0 , which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals:
where sup is the supremum function, which is equivalent to set union in the case of the von Neumann representation of ordinals.