We know that $L$ is the minimal standard model of ZFC.
The question is, what is the minimal "standard" model of ZFC$^-$ (meaning ZFC without the Power Set axiom)?
This is really two questions:
- Is there such a minimal model (this is non-trivial, but sound intuitively true to me)? If so, let's call it $L^-$.
- Assuming $L^-$ exists, is it equal to, or is it a proper subset of $L$ (obviously $L$ itself is a model of ZFC$^-$ so $L^- \subseteq L$)?
Both possible answers for (2) seem puzzling to me. If $L^-=L$ then somehow the Power Set axiom is sufficiently embedded in ZFC$^-$, that even if we don't assume it, it kind of crops up by itself in the minimal model (kind of the way AC and GCH do in $L$).
If $L^-\subsetneq L$ then a whole host of new questions pop up - like whether we have AC if we only assume ZF$^-$ to begin with, and what happens to all the other nice properties $L$ has. Furthermore, if $L^-$ doesn't have the exact same cardinals as $L$, then it appears to me we have some ordinal in $L$ that the power set axiom was necessary to show it has the same cardinality (which sounds peculiar because the bijection has the same cardinality, so why did we "need" anything bigger to construct it? I would have thought the axiom of replacement would be enough).