Take ZFC, remove axiom of Power set, and put instead of it the following axiom:


**Axiom of Successor Cardinals:** $\forall \kappa\, \exists x \, \forall \alpha \, ( \alpha \leq \kappa \to \alpha \in x)$


where "$\leq$" refers to "cardinal smaller than or equal" relation, and $\kappa, \alpha$ range over von Neumann ordinals.


Note: the relation $\leq$ is defined as:


$\alpha \leq \beta \iff \exists f \, (f:\alpha \to \beta \land f \text { is an injection})$


so, it need not be confused with ordinal smaller than or equal relation.


So, the axiom is saying that for every cardinal $\kappa$, there is a successor cardinal $\kappa^+$.



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> Can the resulting theory still interpret ZFC?
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The idea is that if we can develop Gödel's constructible universe **L** inside this system, then this would interpret ZFC? So the power set axiom won't be essential for the development of **L**?



Is the power set axiom essential for constructing L?
KP alone - which is vastly weaker than the theory in question - proves the sentence "For every ordinal $\alpha$, $L\_\alpha$ exists," since it is strong enough to enable effective transfinite recursion. (We're passing to an unnecessarily weak subtheory, but it's worth noting.) The proof of this can be found e.g. in Barwise's book.


The condensation lemma, appropriately stated, can also be proved in KP; since our theory proves that successor cardinals exist, we get powerset in $L$. The proof that $L$ satisfies the rest of the ZFC axioms is the usual one.


So there is a uniform way to define in an arbitrary model $M$ of your theory an inner model (= transitive subclass containing all the ordinals of $M$) which is a model of ZFC + V=L.




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Note the role of condensation in the above: **condensation reduces powersets in $L$ to successor cardinals in $L$ (and hence a fortiori in any larger class)**. So it's not so much that we're avoiding powerset in building $L$, but rather that a very weak theory proves that powerset-in-$L$ is equivalent to successor-cardinals-in-$L$.


