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@TimButton: Does $\mathbf Unbounded$ hold at the first strongly inaccessible cardinal? And do you mention the existence of the first strongly inaccessible cardinal in your published paper?
@AlecRhea: Is it possible to provide us the definitions of $\mathbf Endless$, $\mathbf Infinity$, and $\mathbf Unbounded$? It would help me ( and others) see how these three axioms are related as to the height of this hierarchy. Is there in fact a theorem in the paper you quoted that so relates these three axioms? Thanks in advance for supplying the requested info.
Can one have the following situation: $L_{\infty}$ is countable (and if so then since $L^{M}_{\infty}$ = $M$ if one has that $M$ is a ctm of $ZF$, can one have generic extensions of $L^{M}_{\infty}$)?
