User blog comment:Alemagno12/An extremely fast-growing OCF/@comment-5029411-20170726203426/@comment-30754445-20170727015207

The limit Ω, ΩΩ, ΩΩΩ ... is not I. I is much bigger than that.

The whole point of "inaccessible cardinals" is that they are... well, inaccessible. They cannot be defined by statements of the form "the smallest ordinal not in set X" unless X invokes the inaccessible cardinal itself.

That's why they are so strong. Without the "inaccessible" trick, your system will get stuck at around Ψ(Ψ𝐈(1)) (perhaps a little higher. Depends on how much recursion you'll pack in the rest of the system).