nLab epsilon-number

Definition

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

Properties

The ordinal $\varepsilon$-numbers are unbounded: for every ordinal $\beta$ there is an $\varepsilon$-number greater than $\beta$, namely the limit of the sequence $\beta + 1, \omega^{\beta + 1} ,\omega^{\omega^{\beta + 1}},\dots$. Thus, there are a proper class of ordinal $\varepsilon$-numbers, and their order type? is the same as that of the ordinals themselves. Hence, we can label them by writing $\varepsilon_n$ for the $n^{th}$ $\varepsilon$-number, where $n$ is any ordinal.

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

$$\omega^{\omega^{\omega^{\dots}}}$$

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

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