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Vopenka's principle implies the existence of weakly compact cardinals (a proper class of them, I believe). My question is whether Vopenka's principle is consistent with the assertion that the universe …

Can a Vopenka cardinal be supercompact? I asked a weaker question on here before. Unfortunately, I don't know enough set theory to see whether the positive answer there generalizes to a positive answe …

I believe the model-theoretic version of the question is: is there a theory in finitary first-order logic which has, for each cardinal $\lambda$, a set $C_\lambda$ of $\lambda$-many models, such that …

Here's a question from a non-set-theorist, but a sometime-user of large cardinals. The name Cantor's attic is pretty evocative for the collection of large cardinal axioms: looking through the pages th …

For any given extension $T$ of ZFC (or perhaps NBGC or something), we can ask whether there is an extension $T'$ of ZF which does not prove AC such that $Con(T) \leftrightarrow Con(T')$ $Con(T) \to …

In these slides (see especially slide 26), Steel emphasizes the phenomenon that for all known "natural" extensions of ZFC, the ordering by consistency strength agrees with the ordering by containment …

Let $ZC$ be Zermelo set theory with choice, which differs from $ZFC$ in omitting the axiom scheme of replacement. EDIT: I think I want to include foundation in the axioms, which apparently isn't norma …

Weak Vopěnka's principle says that the opposite of the category of ordinals cannot be fully embedded in any locally presentable category. Recall that one form of Vopěnka's principle says that th …

The question is: if I assert in ZF that there exists a Reinhardt cardinal, do I really get a theory of higher consistency strength than when I assert in ZFC that there exists an I0 cardinal (the stron …

Kunen showed that Reinhardt cardinals are inconsistent in ZFC. But his proof is a bit technical for a non-set-theorist to follow. Berkeley cardinals are stronger than Reinhardt cardinals. You can refu …

Let $\kappa$ be an inaccessible cardinal. Recall that $\kappa$ is weakly compact if every tree of height $\kappa$ has either a level of size $\kappa$ or a branch of size $\kappa$. So if $\kappa$ is a …

Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with …

I am aware of three major "hierarchies" of mathematical theories, but I don't know how to relate these hierarchies to one another. Here are the hierarchies I have in mind: Consistency strength. My u …