Suppose $ZFC^-$ is $ZFC$ minus the power set axiom, and that, for $\gamma$ a countable ordinal, $\mathcal{P}^\gamma$ is an axiom that allows less than $\gamma$ applications of the power set operation. Let S be $ZFC^-$ plus $\mathcal{P}^\gamma$. Is there a countable ordinal $\alpha$ so that $L_\alpha$ is a model of S if S has a countable model?