I am trying to prove that $ \omega_1$ is strongly inaccessible in $ L(r)$ for all $ r \subseteq \omega$ from the assumption that $ \omega_1$ is inaccessible to reals, i.e., that $ \omega_1^{L(r)} < \omega_1^V$ for all $ r \subseteq \omega$ .

I have managed to prove that $ \omega_1^V$ is an uncountable regular cardinal in $ L(r)$ , but failed at proving it is also a strong limit.

The class $ L(r)$ is defined by transfinite recursion as follows:

• $ L_0(r) = \mathsf{trcl}(r)$ ;
• $ L_{\alpha+1}(r) = D^{+}(L_\alpha(r))$ ;
• $ L_\lambda(r) = \bigcup_{\alpha < \lambda} L_\alpha(r)$ , if $ \lambda$ is a limit ordinal.
• $ L(r) = \bigcup_{\alpha \in \mathsf{Ord}} L_\alpha(x)$ .

The set $ \mathsf{trcl}(A)$ is the transitive closure of $ A$ , and $ D^{+}(A)$ is the set of all sets definable in $ A$ with or without parameters in $ A$ .