Define $\kappa$ as $\Sigma_n$-weakly berkeley cardinal if for any transitive set $M$ that includes $\kappa$ exist elementary embedding $j:M\rightarrow M$ save only $\Sigma_n$ formulas and critical point below $\kappa$. $\prod_n$-weakly berkeley cardinal define similarly. This cardinals consistent with $ZF(C)$?
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1$\begingroup$ If I understand you correctly, a $\Pi_1$-weakly Berkeley cardinal is (proto)Berkeley (and probably $\Sigma_0$ already suffices). Hint: if $j : V_{\alpha+1}\to V_{\alpha+1}$ is $\Pi_1$-elementary, then $j\restriction V_\alpha$ is a fully embedding from $V_\alpha$ to $V_\alpha$. $\endgroup$– Gabe GoldbergCommented Aug 25, 2020 at 23:15
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$\begingroup$ @Gabe: So you can take $j\colon V_{\alpha+2}\to V_{\alpha+2}$ to be $\Sigma_0$-elementary to get a fully elementary $j\colon V_\alpha\to V_\alpha$? $\endgroup$– Asaf Karagila ♦Commented Aug 26, 2020 at 0:39
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$\begingroup$ @AsafKaragila Well you need to know that $j(V_{\alpha+1}) = V_{\alpha+1}$ which doesn't follow (since any $j : V\to M$ restricts to $\Sigma_0$ embeddings from $V_\alpha$ to $V_\alpha$ for all $\alpha$ with $j[\alpha]\subseteq \alpha$). But you can probably modify the transitive set... I didn't want to think about it. $\endgroup$– Gabe GoldbergCommented Aug 26, 2020 at 0:51
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