(This question was posted on math.SE over two weeks ago, but received no answer. I am therefore posting it here as well.) It is well-known that there are difficulties in developin …

A cardinal $\theta$ is worldly if $V_{\theta}$ is a model of ZFC. We could force to collapse $\theta$ to a successor cardinal, for example, and destroy the worldliness of $\theta$ …

Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? A …

Is it consistent with ZFC to have a cardinal $\kappa$ which is not Ramsey and $\kappa \rightarrow [\kappa]^{<\omega}_{\omega,n}$ holds for some $n\in \omega$? The partition rel …

There is a tension between (1) interpreting proper class talk in set theory as talk about first-order formulas and satisfaction; and (2) taking it to be an interesting and non-triv …

This is a question in two parts about the interaction of surreal numbers and large cardinals, in both cases just a request for references on the subject. Part 1 is about foundatio …

I would like to suggest another argument for Kunen's inconsistency result, and I wonder to know if the argument is correct. I am also interested to see, if the proof is correct, wh …

In chapter 8 of Mitchell's and Steel's FSIT, they prove a central fine structural result, which basically states that if $\mathcal{M}$ is 1-small, $k$-sound, $k$-iterable premouse …

Let $L_3$ be the third-order language of set theory with identity on the first sort. Variables $x$ are first-order, $y$ are second-order, and $z$ are third-order. In the style of L …

Is anything known about the consistency strength of the following statement? $\kappa$ is a Mahlo cardinal and there is a stationary set of $a \in \mathcal{P}_\kappa(\kappa^+)$ su …

If $r, s\in\mathbb{R}$, we say $r$ is constructible relative to $s$ - and write $r\le_cs$ - if $r\in L[s]$. Modding out by the induced equivalence relation $\equiv_c$, we get a par …

Today we have that $L(\mathbb{R}) \models AD$ (assuming there are $\omega$ many Woodin cardinals and a measurable above them all). I was wondering what other models of $AD$ might l …

Let $\mathcal M$ be a premouse. $\mathcal M$ is said to be $\omega$-small if and only if whenever we have that $\kappa=crit(E)$ for some extender $E$ on the sequence of $\mathcal M …

A well-known result of Kunen says that there is no non-trivial elementary embedding $j: V \rightarrow V.$ There are several proofs of this theorem (see Kanamori, The higher infinit …

Does anyone have a reference for the consistency strength of saying that the Erdos cardinal $\kappa(\alpha)$ exists for all $\alpha$? Or of various weakenings about how far such c …