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Here is a characterization of $\lambda$, but of course your question is somewhat ambiguous as to what counts as an answer. For $n<\omega$, let $\alpha_n$ be the least ordinal $\alpha$ such that $\kapp …

No. Under the hypotheses, there are limit ordinals $\eta$ such that $\mathcal{P}(\eta)\cap L\subseteq\mathfrak{L}_{\eta+1}$, and therefore, for example, $L_{\eta+2}\cap\mathfrak{L}_{\eta+1}\neq L_{\et …

To summarize the comments, the hypothesis is equiconsistent with (ZFC +) $0^\sharp$ exists + "there is a worldly cardinal". For as mentioned in the original post, under the hypothesis, $0^\sharp$ exi …

(This answers what was the original question, which was with $\mathrm{Coll}(\omega,{<\kappa})$ replacing $\mathrm{Coll}(\omega,\kappa)$.) Hmm...doesn't this contradict Theorem 1.22 of "MICE WITH FINIT …

Yes. In fact for all ordinals $\xi\notin\mathrm{range}(j_0)$, we have $j_1(\xi)<j_0(\xi)$. For let $\xi$ be an ordinal such that $\xi\notin\mathrm{range}(j_0)$. Let $\gamma$ be least such that $j_0(\g …

Here is an upper bound: Suppose $\kappa$ is $2$-fold supercompact. Then the property holds at $\kappa$. (Recall that $2$-fold supcompactness means that for each ordinal $\lambda$, there is $j:V\to M$ …

If there is a normal measure $U$ on $\kappa$ such that $A_X\in U$, where $A_X$ is the set of $X$-cardinals $\alpha<\kappa$, then not only is $\kappa$ a limit of inaccessiles $\kappa'$ such that $\kapp …

Question 1: Yes, in fact if $M\models$ZFC+"There is a transitive model of $T$" where $T$ is the theory ZFC + "There are two distinct inaccessibles", then there is such an $(A,\alpha)$ - just let $A$ b …

(EDIT: Now edited to compute the precise value of $\alpha$.) @AsafKaragila already answered the question, but this is answering the follow-up question od @Reflecting_Ordinal in the comments to Asaf's …

Yes. The elements of $L_{\alpha+1}$ are exactly those subsets of $L_\alpha$ which are definable from parameters over $L_\alpha$. But $L_\alpha\models\mathrm{ZFC}$, so from here we can just use the usu …

Yes. And if the $\mathscr{I}_k$s are normal, then the suggested candidate works (I haven't really thought about whether the candidate still works without normality). Here is the argument, which is a t …

Remarks: (i) I'm interpreting the definition of $\leq_{L,\mathrm{end}}$ as quantifying over set models $W'$, not proper classes. (ii) I'm considering the main question (comparing the two orders), par …

(This should fix an earlier attempt...) I don't see how one can prove that $V$ is the union of that particular chain, but here is a variant, from which we still get finite axiomatizability. Suppose f …

Much as described by @ElliotGlazer above, let's consider the question for models of the form $(V_\kappa,V_{\kappa+1})$, assuming ZFC in the background, considering all such where $\kappa$ is inaccessi …

Theorem 1. If $V$ is a non-trivial set generic extension of $W\models\mathrm{ZFC}$ then there is no $j:V\to W$ as described (i.e. with $x\in y\iff j(x)\in j(y)$ for all $x,y\in V$). (So in particular, …