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Here's a counterexample for $\kappa=\aleph_1$: let $B$ be the structure with underlying set $\mathbb{N}\sqcup\mathcal{P}(\mathbb{N})$, equipped with the usual ordering on $\mathbb{N}$ as well as the $ …

Unless I'm missing something, if $\vert T\vert+\aleph_0<\kappa$ and $\kappa^+=2^\kappa$, then we can build a saturated model of $T$ of cardinality $2^\kappa$. So: If we want a countable theory with …

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 …

Sorry this is a bit disjointed - there's a lot of stuff here. I hope this helps though. All of these notions are applicable in all contexts - or at least, all sufficiently rich contexts (we probabl …

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 …

Perhaps surprisingly, the Open Coloring Axiom (even OCA + MA$_{\aleph_1}$) has no additional consistency strength. The situation is described in Velickovic's paper "Applications of the open coloring a …

This is a counterexample to Vopenka's principle phrased slightly differently: as "in any proper class of first-order structures, one elementarily embeds into the other." Working in $V=L$, I claim tha …

Here's a brief sketch of why, assuming $ZFC+I_0$ is consistent, so is $ZFC+CH+I_0$. (This is just Levy-Solovay.) Suppose $\lambda$ is $I_0$ - that is, there is a nontrivial elementary embedding $j$ o …

The truth of a first-order sentence $\varphi$ in a structure $\mathfrak{M}$ is absolute between $V$ (= reality) and sufficiently large transitive sets containing $\mathfrak{M}$. In particular, already …

Re: 6: You're right, Friedman certainly has something to say on this topic! Friedman has discovered a number of $\Pi^0_1$ sentences of Ramseyish flavor which have large cardinal strength. For example, …

Let me give an argument, not that we should believe large cardinal axioms or their consistency, but rather that regardless of our belief in consistency we should still be interested in results around …

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 …

I emailed Itay Neeman, and he told me the following: As far as I know it's open. I don't think anything is known about unraveling beyond what you can get from my methods. These give the Suslin operat …

Suppose we have an elementary embedding $j: V\rightarrow V[G]$; why should we expect $ran(j)\subset V$? This would only need to be true if $V$ were definable in $V[G]$. Now, by a theorem of Laver (and …

Not really a complete answer, but too long for a comment: If I understand you correctly, the answer is yes, this idea exists in multiple different forms. The one I find most intriguing currently is J …