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_\...