A gap in the constructible universe $L$ occurs at an ordinal $\beta$ if $L_\beta\cap P(\mathbb{N})=L_{\beta+1}\cap P(\mathbb{N})$. Now $ZFC$ implies that the first gap in the constructible universe occurs at a large nonrecursive ordinal $\beta_0$, known as the ordinal of ramified analysis. But I’m wondering what it takes to prove that the first gap doesn’t occur much earlier, and that too a longer gap.
My question is, what is the weakest subsystem of $ZFC$ that proves that $L_{\omega+1}\cap P(\mathbb{N})\neq L_{\omega+\omega}\cap P(\mathbb{N})$? That is, that a gap in the constructible universe of a length $\omega$ does not occur at $\omega+1$. Or to put it another way, that there exists a natural number $n$ such that $(L_{\omega+n}-L_{\omega+1})\cap P(\mathbb{N})\neq\varnothing$.
Would Kripke-Platek Set Theory, Zermelo Set Theory, or something weaker suffice? In any case, the motivation of my question is that this is basically the negation of Bertrand Russell’s Axiom of Reducibility.