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I think the answer is no. If $\kappa$ is huge, and we derive an almost-huge tower from the huge embedding, then the embedding computed from the tower has stationary correctness. Suppose $h : V \to M …

This isn't really an answer, but here's a way you can get the maps as you want with the embeddings defined outside of the model. Assume $0^\sharp$ exists, and let $\langle \alpha_i : i < \omega^2 \ra …

Referring to proper classes by means of constants clearly doesn’t use the “internal properties” of the universe of sets. You’re literally reaching outside the universe and adding an artificial means …

Yes, if $F$ is large enough. Let $j : V \to M$ be the embedding derived from an $\aleph_1$-strongly compact ultrafilter $U$ on $P_\kappa(\lambda)$. Let $[\mathrm{id}]$ be the set in $M$ represented …

First let us show that consistently there is no such bound, along with $2^\kappa$ larger than any prescribed cardinal. Assume $\diamondsuit_\kappa$. This is consistent with any large cardinal assumpt …

I'm not sure if this gets to the heart of the question, but here are some observations. For a successor ordinal $\alpha$, let $\kappa_\alpha$ be the least $\kappa>\alpha$ which is $\kappa^{+\alpha}$-s …

I think it can be reasonably argued that the large cardinal notions follow a common conceptual pattern, even a semi-formal template. Once we understood the characterization of measurable cardinals in …

We now have a full answer. The capturing property is equivalent to weak compactness. Yair already covered the non-inaccessible case. Now we show that this property implies the tree property at $\ka …

A sketch of Silver's original argument appears in section 19 here: http://math.bu.edu/people/aki/e.pdf

This question is more subtle than I originally thought. The answer is that the theory is consistent assuming some large cardinal hypothesis, the existence of $0^\sharp$. If $0^\sharp$ exists, then t …

This is an instance of a more general phenomenon. Let $\kappa$ be a regular cardinal, and $I$ be a $\kappa$-complete ideal on some set $Z$. Let $\mathbb{P}$ be a forcing of size less than $\kappa$, …

For each $\alpha<\omega_1$, choose a real $r_\alpha$ that codes the ordertype $\alpha$. This sequence $\langle r_\alpha : \alpha < \omega_1 \rangle$ then codes a sequence of surjections from $\omega$ …

The statement implies that for any $f : [\omega_1]^2 \to 2$, there is an uncountable $S$ such that $f$ is constant on $[S]^2$. This is impossible since $\omega_1$ is not weakly compact. See Jech, Le …

I suppose there is some interest in propositions implying large powerset. For example, a classical question at the infancy of set theory and real analysis was whether there is a probability measure on …

To complement Joel's answer and tie into the issue of $L$, consider a model $L[U]$, where $U$ is a normal measure on a cardinal $\kappa$. It is well-known that $L[U]$ satisfies GCH and $\diamondsuit_ …