Epsilon number

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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

${\displaystyle \varepsilon =\omega ^{\varepsilon },\,}$

in which ω is the smallest infinite ordinal.

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

${\displaystyle \varepsilon _{0}=\omega ^{\omega ^{\omega ^{\cdot ^{\cdot ^{\cdot }}}}}=\sup \left\{\omega ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\omega ^{\omega ^{\omega ^{\omega }}},\dots \right\}\,,}$

where sup is the supremum, which is equivalent to set union in the case of the von Neumann representation of ordinals.

Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in ${\displaystyle \varepsilon _{1},\varepsilon _{2},\ldots ,\varepsilon _{\omega },\varepsilon _{\omega +1},\ldots ,\varepsilon _{\varepsilon _{0}},\ldots ,\varepsilon _{\varepsilon _{1}},\ldots ,\varepsilon _{\varepsilon _{\varepsilon _{\cdot _{\cdot _{\cdot }}}}},\ldots \zeta _{0}=\varphi _{2}(0)}$.^[1] The ordinal ε₀ is still countable, as is any epsilon number whose index is countable. Uncountable ordinals also exist, along with uncountable epsilon numbers whose index is an uncountable ordinal.

The smallest epsilon number ε₀ appears in many induction proofs, because for many purposes transfinite induction is only required up to ε₀ (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness theorem, show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with this property, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peano arithmetic).

Many larger epsilon numbers can be defined using the Veblen function.

A more general class of epsilon numbers has been identified by John Horton Conway and Donald Knuth in the surreal number system, consisting of all surreals that are fixed points of the base ω exponential map x → ω^x.

Hessenberg (1906) defined gamma numbers (see additively indecomposable ordinal) to be numbers γ > 0 such that α + γ = γ whenever α < γ, and delta numbers (see multiplicatively indecomposable ordinal) to be numbers δ > 1 such that αδ = δ whenever 0 < α < δ, and epsilon numbers to be numbers ε > 2 such that α^ε = ε whenever 1 < α < ε. His gamma numbers are those of the form ω^β, and his delta numbers are those of the form ω^ωβ.