nLab
epsilon-number

ε\varepsilon-numbers

Definition

An ε\varepsilon-number is an ordinal number, or more generally a surreal number, α\alpha, such that α=ω α\alpha = \omega^\alpha, where ω\omega is the first infinite ordinal. In other words, it is a fixed point of the exponential map λx.ω x\lambda x. \omega^x.

Properties

The ordinal ε\varepsilon-numbers are unbounded: for every ordinal β\beta there is an ε\varepsilon-number greater than β\beta, namely the limit of the sequence β,ω β,ω ω β,\beta, \omega^\beta ,\omega^{\omega^\beta},\dots. Thus, there are a proper class of ordinal ε\varepsilon-numbers, and their order type? is the same as that of the ordinals themselves. Hence, we can label them by writing ε n\varepsilon_n for the n thn^{th} ε\varepsilon-number, where nn is any ordinal.

In particular, ε 0\varepsilon_0 is the first ordinal ε\varepsilon-number, namely the limit of the sequence 1,ω,ω ω,ω ω ω,1,\omega, \omega^\omega ,\omega^{\omega^\omega},\dots. It may be tempting to write ε 0\varepsilon_0 as

ω ω ω \omega^{\omega^{\omega^{\dots}}}

but in fact every ε\varepsilon-number could be written in this way, so this notation does not specify a unique number.

Similarly, the surreal ε\varepsilon-numbers have the same order type as the surreals, and can be notated as ε n\varepsilon_n where nn is a surreal.

Created on January 18, 2016 at 21:22:03. See the history of this page for a list of all contributions to it.