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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 $j:V\to M$ …

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 …

(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 …

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, …

(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 …

Complementing @JasonChen's answer: Assume ZFC+$I_1$ and let $j:V_{\lambda+1}\to V_{\lambda+1}$ be elementary, so $\lambda$ is the sup of the critical sequence of $j$. Then $V_{\lambda}$ models $\mathf …

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 …

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 …

Some remarks: By Schindler's paper "Iterates of the core model", if $j:V\to N$ is elementary ($N$ transitive) and $N$ is closed under $\omega$-sequences, and $k:K\to K^N$ is the restriction of $j$, th …

I think that if you have a logic $\mathcal{L}$ which has downward Lowenheim-Skolem (for theories of arbitrary cardinality, i.e. if $T$ has cardinality $\lambda$ and $N$ is a model $A$ of size $\theta\ …

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 …

Note that $M=\bigcap_{\alpha}M_\alpha$ is just the $\mathrm{OR}^{\mathrm{th}}$ iterate $M_{\mathrm{OR}}$ cut off at height $\mathrm{OR}$, so we have for example $V_\kappa\preceq V_\lambda\preceq M$ wh …

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 …

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 …

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 …