# What's the consistency strength of this kind of cardinal?

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?

• 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.) May 13, 2022 at 17:21
• Oh. I double-checked my friend's message and I do have my arrow backwards apparently. May 13, 2022 at 17:35
• What is the motivation for this definition? What is $X$? May 17, 2022 at 9:13
• X is a class. Also, I guess that this is a "nonrecursive analogue" of stability. May 17, 2022 at 18:31