I'm interested in the existence and properties of an analogue version of $L$ for models of ZF$^-$ (ZF without the power set axiom), which for simplicity I'll call $L^-$. By "analogue" I mean the least inner model of a set theory universe $V$ (same ordinals and transitive).
- Assuming $V$ is a model of ZF$^-$, does a least inner model exist, and is it unique? Intuitively I'd answer that as "yes", though I don't know how to prove it.
- Assuming "yes" for question 1, and assuming $V$ is a model of ZF (note here that $V$ does satisfy the power set axiom), is $L^- = L$ (obviously $L$ itself is a model of ZF$^-$, so $L^- \subseteq L$)?
Assuming existence and uniqueness, I strongly suspect equality, though it isn't obvious to me. If we look at the power set axiom as a large cardinal axiom (postulating the existence of some large cardinal with a certain property), then this just means that $L$ "inherits" these "large cardinals" from the universe. On the other hand, if $L^-$ is actually a proper subclass of $L$, then a host of new questions about this $L^-$ spring to mind.
- In a universe where $V$ is a model of ZF$^-$ only, does $L^-$ satisfy AC? This question seems to be a little bit trickier than for $L$, because of our non-constructive definition of $L^-$.