Let $\beta$ be a (non-empty) ordinal.
Is $0^\beta = 0?$ Or is it $1$?
My definition is the following: let $\alpha, \beta$ ordinals
$\alpha^0= 1$, $\alpha^{\beta+1} = \alpha^\beta \alpha, \alpha^\beta = \bigcup_{\eta < \beta} \alpha^\eta$ if $\beta$ is a non-zero limit ordinal.
From this definition, it would seem that $0^0 = 1$, $0^1 = 1. 0=0, \dots$
$$0^\omega= \bigcup_{\eta <\omega}\alpha^\eta= 1$$???
These results don't seem to make sense.