User blog comment:Emlightened/Early Birthday Present For Deedlit/@comment-1605058-20180304163730

I've looked at your 2) and I think there might be two problems with it. I am not too knowledgeable about forcing, so apologies if what I say makes no sense.

Firstly, as far as I know, there is no such thing as the Levy collapse (of one cardinal to the other). Indeed, everything is going to depend on what generic set you take, and different generic sets can (within my knowledge) lead to non-isomorphic or even non-equivalent extensions, so whatever definition you make using Levy collapses may depend on arbitrary choices involved in the construction of the extension.

Secondly, even if we do assume we have fixed Levy collapses for all cardinal (of interest), I believe the definition might not do what you want it to do. Indeed, you consider countable ordinals \(\beta\), and then you consider the value of \(o(\beta)\) in \(V[G_{\Omega\leftarrow|\alpha|^+}]\) (I assume this is what you mean with \((o(\beta)^{V[G_{\Omega\leftarrow|\alpha|^+}]}\)). However, \(\beta\) is still countable in every generic extension (since we don't remove the bijection between \(\beta\) and \(\omega\), we only add things). On the other hand, \(o(\beta)\) trivializes for countable ordinals - you define it by considering all sets of regular ordinals with supremum equal to \(\beta\), but the only regular ordinals below it are \(0,1,\omega\), and except for a few very isolated cases, we won't be able to have supremum \(\beta\), so \(o(\beta)=0\) in all extensions, so it will hardly ever be \(\geq\alpha\).