Epsilon zero

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. Any solution to this equation has Cantor normal form $\varepsilon =\omega ^{\varepsilon }$ $\varepsilon =\omega ^{\varepsilon }$ .

The least such ordinal is ε₀ (pronounced epsilon nought or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals:

Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in $\varepsilon _{1},\varepsilon _{2},\ldots ,\varepsilon _{\omega },\varepsilon _{\omega +1},\ldots ,\varepsilon _{\varepsilon _{0}},\ldots ,\varepsilon _{\varepsilon _{1}},\ldots ,\varepsilon _{\varepsilon _{\varepsilon _{\cdot _{\cdot _{\cdot }}}}},\ldots$ $\varepsilon _{1},\varepsilon _{2},\ldots ,\varepsilon _{\omega },\varepsilon _{\omega +1},\ldots ,\varepsilon _{\varepsilon _{0}},\ldots ,\varepsilon _{\varepsilon _{1}},\ldots ,\varepsilon _{\varepsilon _{\varepsilon _{\cdot _{\cdot _{\cdot }}}}},\ldots$ . The ordinal ε₀ is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals, and uncountable epsilon numbers whose index is an uncountable ordinal).

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