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I believe the answer is "yes" - $V_\lambda$ can compute $\varphi^{V_\kappa}$ for every formula $\varphi$ (since the full truth predicate for $V_\kappa$ lives already at around $V_{\kappa+2}$ or so), a …

A $\Pi_1$ statement $\varphi$ is absolute downwards between transitive structures: if $(V_\alpha; \in, A)\models\varphi$ and $\beta<\alpha$, then $(V_\beta;\in, A\cap V_\beta)\models\varphi$. So $\Pi^ …

EDIT: Based on the comments, I think it's worth clarifying a bit of the nature of the Boffa-Jensen construction of a model of NFU (which appears to be part of the motivation for this question). In th …

There is no contradiction here. Look at Theorem $2.3$: Suppose $I\subseteq On$ witnesses $On$ is Ramsey. Then, definably over $\langle V,\in, I\rangle$, there is a transitive class $M$, and an …

This is extremely sketchy since I have to run, but I'll fill in the details later: I think if $j$ comes from a measurable - more generally, if $M^\kappa\subseteq M$ - then we already have $\mathcal{L …

Since this is really just a question about absoluteness, let me leave inaccessibility behind for the moment and focus on a simpler still-interesting example: cardinal-ness. Suppose $\alpha$ is an ord …

I missed the inaccessibility requirement initially - fixed! Since $\mathsf{TG}$ is "just" $\mathsf{ZFC}$ + "There is a proper class of inaccessible cardinals," an inaccessible $\alpha$ satisfies $V_\a …

Indeed, you cannot have an elementary embedding $j:V\rightarrow M$ with critical point $\omega_1^V$. However, there are some subtleties to keep in mind: We could have an elementary embedding $j:W\ri …

Let me argue that Kunen's argument actually shows the best possible thing here. First, let's think about consistency results. The "ideal" result here would be: (i) Con(ZFC) implies Con(ZFC + "The …

Two comments on this: First, it's not clear that one can formulate "there exists a transitive set $S\in V$ such that $S\prec V$" in first-order logic, so it's a bit tricky to phrase your question pre …

Really, this was answered in the comments; I'm putting this answer down to move this off the unanswered queue. I've made this CW and will delete it if one of the original commenters adds their own ans …

Dropping transitivity doesn't actually add anything: given $M\supseteq\kappa$, just consider $\hat{M}=$ the Mostowski collapse of $M$. Given $\alpha<\kappa$ and $j:\hat{M}\rightarrow \hat{M}$ nontrivi …

This question can be more clearly phrased as: Is the principle "We can iterate powerset along any (definable) class-well-ordering" (which is really a scheme, appropriately) consistent with ZFC? …

It depends on the large cardinal property. For example, inaccessibility is downwards-absolute: if $V\models$ "$\kappa$ is inaccessible" and $W$ is an inner model of $V$ then $W\models$ "$\kappa$ is in …

At least for Mahlo-ness, things are pretty simple to describe: Given a property $P(\kappa)$ of cardinals implying strong inaccessibility, let $P^{\mathrm{Mahlo}}(\kappa)$ be the property "$\kappa$ ha …