# Talk:Huge

## $\omega$-almost hugeness underexplored?

The page briefly mentions the $\omega$-almost huge cardinals, but then immediately moves on to $\omega$-huge cardinals without mentioning anything about the properties or possibility/impossibility of $\omega$-almost huge cardinals. Furthermore, none of the other pages in this wiki make any mention of $\omega$-almost hugeness. Is it consistent for them to exist, how strong are they, and is it known how they relate to other large cardinals? Eaglgenes101 (talk) 14:23, 27 April 2019 (PDT)

So I thought about it a bit. The first fixed point of the elementary embedding $j$ above $crit(j)$ is the supremum of all of $crit(j)$, $j(crit(j))$, $j^2(crit(j))$, $j^3(crit(j))$, etc..., and said iterates of $j$ form a cofinal set of $j^\omega(crit(j))$. So for any $\alpha < j^\omega(crit(j))$, there is some whole number $n$ such that $j^n(crit(j)) > \alpha$. If I reasoned correctly, this means that $\omega$-almost hugeness is equivalent to $n$-hugeness for all whole number $n$, which is known to be perfectly consistent at least relative to an $I3$ cardinal. Eaglgenes101 (talk) 08:26, 29 April 2019 (PDT)