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This answer will answer question 1 and almost answer question 2, giving a positive answer to question 2 when $\beta$ is a limit ordinal. The linked paper cites theorem 6.1 of Schmerl's "On $\kappa$-li …

There is a characterization of $\alpha$-Erdős cardinals in terms of sets of indiscernibles of order type $\alpha$. There is also a characterization of Ramsey cardinals in terms of sets of good indisce …

In discussion about the consistency of large cardinals, a justification which occasionally appears is that despite years of work, no inconsistency has currently been discovered. For example, here are …

Are there cardinal axioms $A$ for which KP+$A$ or CZF+$A$ are as strong as ZFC-(Powerset axiom)+$A$? How about ZF+$A$? In this answer I am going to treat this as a "know it when you see it" question …

On December 13--20 2009 at Bristol, there was a meeting devoted to thorough dissection of Harvey Friedman's work on the foundations of mathematics and his statements claimed to be equivalent to consis …

A cardinal is Reinhardt if $\kappa$ is the critical point of a nontrivial elementary embedding of $V$ to itself, where $V$ is the class of all sets. As Reinhardt cardinals are inconsistent with $\math …

Friedman has made public on his website a 2016 draft titled "Expanding Mind Theory". On page 4 there is a definition of a theory $\mathrm{EM}$ in first-order logic which formalizes the theory of two m …

In "On the consistency of ZF with an elementary embedding from $V_{\lambda+2}$ into $V_{\lambda+2}$" (arXiv 2006.01077, 2020) Farmer Schlutzenberg showed that assuming a theory based on $I0$ and some …

On page 241 of William Reinhardt's paper "Ackermann's set theory equals ZF" (Annals of Math. Logic vol. 2, 1970), question 4.15 is the following: How large is the first ordinal $\gamma$ such that the …

For about two years I have been in contact with a splinter group of amateur enthusiasts who have been striving towards this endeavor. One of their considered objects is "an infinity so large, that it …

The Hyperuniverse Program, founded by Sy D. Friedman, intends to produce new second-order axioms of set theory which appropriately formalize "the universe is maximal" in one of a few ways. A height-ma …