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Not necessarily. Here's a counterexample, but I'm sure it is a ridiculous overkill in consistency strength: Suppose $\kappa$ is the least inaccessible limit of supercompact cardinals. Then $\kappa$ …

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 …

I believe it is fair to say that the HOD Conjecture has been refuted. The HOD Conjecture is the statement that the theory ZFC + "There is an extendible cardinal" proves that there is a proper class of …

Real-valued measurable cardinals (RVM) are equiconsistent with 2-valued measurable cardinals. I believe this is due to Solovay and Kunen. (Solovay for the forcing direction, Kunen for the inner mode …

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 …

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 …

First, your notation is nonstandard; when we write $\mathrm{Col}(\kappa,\lambda)$, this typically means the set of partial functions $p : \kappa \to \lambda$ of size $<\kappa$, i.e. reverse of yours. …

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_ …

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

If we start with an indestructibly supercompact cardinal $\kappa$, then we can do Easton forcing to get any reasonable continuum function on the regular cardinals $\geq \kappa$ and retain the supercom …

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 $\omega$ …

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 …

For question 2, some large cardinals imply the existence of such a forcing. Suppose $\kappa$ is 2-huge, with $j : V \to M$ an elementary embedding, $\lambda = j(\kappa)$, $\theta = j(\lambda)$, and $ …

I'll give the proof, which was told to me by Martin Zeman. Let $G \subseteq \mathrm{Col}(\omega_1,{<}\kappa)$ be generic, where $\kappa$ is Mahlo. Suppose towards contradiction that $\square_{\omega_ …

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 …