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Bumped, since I was recently thinking about these again.

My friend introduced the following notion:

Let $\xi > 0$, $\eta$ be ordinals, $n$ be a natural number and $\mathcal{A}, X$ be classes. A cardinal $\kappa$ is called $\mathcal{A}\textrm{-}\eta\textrm{-}(n,\xi)$-weird on $X$ iff there is a $\Sigma^1_{2n}$-elementary embedding $j: \langle V_{\kappa+\eta}; \in, \mathcal{A} \cap V_{\kappa+\eta} \rangle \to \langle V_{\kappa+\eta+\xi}; \in, \mathcal{A} \cap V_{\kappa+\eta+\xi} \rangle $ with critical point $\kappa$ and $j(\kappa) \in X$.

How high in consistency strength is it?

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  • $\begingroup$ Do you have your arrow backwards? There already can't be any $\Sigma_0$-elementary embedding from $V_\alpha$ to $V_\beta$ if $\alpha>\beta$. (It would induce an order-embedding $\alpha\rightarrow\beta$, which can't happen by well-orderedness.) $\endgroup$ May 13, 2022 at 17:21
  • $\begingroup$ Oh. I double-checked my friend's message and I do have my arrow backwards apparently. $\endgroup$
    – Binary198
    May 13, 2022 at 17:35
  • $\begingroup$ What is the motivation for this definition? What is $X$? $\endgroup$
    – Yair Hayut
    May 17, 2022 at 9:13
  • $\begingroup$ X is a class. Also, I guess that this is a "nonrecursive analogue" of stability. $\endgroup$
    – Binary198
    May 17, 2022 at 18:31

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