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As I pointed out in the meta thread, this question overlaps with a bunch of older MO questions. Arguments against large cardinals Nonessential use of large cardinals Inaccessible cardinals and Andre …

This product forcing (known as the Lévy collapse) is not ${<\kappa}$-closed for any $\kappa > \omega$ since the individual factors $P_\alpha$ are not ${<\kappa}$-closed. The forcing is $\lambda$-c.c., …

This is a very interesting question. Here is what Shelah says about this in The Future of Set Theory: ISSUE: Where does the truth lie between the following two extremes Every combinatorial statement …

The analogy between universes in type theory and the Mahlo hierarchy in set theory has been analyzed in many different ways by Michael Rathjen. (This builds on his analysis of KPM, but ML type theorie …

It is very near the bottom of Kanamori's chart. The very bottom of the chart is the level of a (strongly) inaccessible cardinals, which is the smallest large cardinal axiom. Right above the inaccessib …

In general, one cannot force the failure of $\square(\kappa)$ at a fixed cardinal $\kappa$. Indeed, if $\kappa$ is any regular uncountable cardinal which is not weakly compact in $L$, then there is a …

The following fact has been called "Ramsey's Theorem for Analysts" by H. P. Rosenthal. Theorem. Let $(a_{i,j})_{i,j=0}^\infty$ be an infinite matrix of real numbers such that $a_i = {\displaystyle …

Here is an extension of Barwise's theorem which may be of some use. Theorem. Fix a real $a \subseteq \omega$ in $W$. Suppose the preorder $\preceq$ is first-order definable with parameter $a$ and that …

It should be noted that Petr Vopěnka himself did not believe in the principle! Here is the story, taken from Adámek and Rosický Locally Presentable and Accessible Categories (p. 278-279). The stor …