# Choice function for elementary embedding $j: V_\lambda\prec V_\lambda$

Work in ZF. Let $$\lambda$$ be strongly inaccessible. Let $$\kappa$$ be such that for every $$\alpha\lt\lambda$$, there is some $$j: V_\lambda\prec V_\lambda$$, with critical point $$\kappa$$, such that $$j(\kappa)\gt\alpha$$.

In this setting, can we define a sequence $$A=\langle j_\alpha: V_\lambda\prec V_\lambda\mid \alpha\lt\lambda\rangle$$, such that every $$j_\alpha$$ is a non-trivial elementary embedding with critical point $$\kappa$$, and $$j_\alpha(\kappa)\gt\alpha$$?

(Note: $$A$$ needs to be a sequence, not just the set of all embeddings.)

Motivation: I was doing research on large cardinals without choice, and was attempting to use a similar collection of embedding to $$A$$ to give a first order definition of cardinals that were $$n$$-huge.

• Can you show in ZF that the set $\{ j(cr(j)) | j : V_\lambda \to V_\lambda$ is elementary$\}$ is cofinal in $\lambda$, assuming it’s nonempty? Oct 30, 2020 at 8:20
• That is the definition of the problem. For any $\alpha$, we have that there is some $j: V_\lambda\prec V_\lambda$ with $j(crit(j))\gt\alpha$. Oct 30, 2020 at 16:23
• If $\lambda$ is the least inaccessible for which there is such an elementary embedding, then that set is not cofinal since $\lambda$ is not a limit of inaccessibles. Oct 30, 2020 at 17:26
• You're right. I assumed he was talking about this specific problem. Oct 30, 2020 at 18:00
• May we assume AC in $V_\kappa$ (and therefore also in $V_\lambda$)? Oct 31, 2020 at 21:25