nLab epsilon-number




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.


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+1,ω β+1,ω ω β+1,\beta + 1, \omega^{\beta + 1} ,\omega^{\omega^{\beta + 1}},\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.

Last revised on March 30, 2024 at 12:40:41. See the history of this page for a list of all contributions to it.