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The following may not be an answer to your question, but I think it is related. I have taken it from the introduction of a joint work I am doing with James Cummings and Sy Friedman (which has now appe …

I am looking for a reference for the following result: Theorem Assume $\kappa$ is a Woodin cardinal. Then after forcing with the Levy collapse $\mathbb{P}=Col(\omega, \kappa)$, the $\Sigma^1_2$-deter …

In the Foreman-Woodin model The generalized continuum hypothesis can fail everywhere. for each infinite cardinal $\kappa, 2^\kappa$ is weakly inaccessible. This answers your last question. The answer …

Stanley, M. C., A cardinal preserving immune partition of the ordinals, Fundam. Math. 148, No. 3, 199-221 (1995). ZBL0843.03028. An infinite set (or class) of ordinals is said to be immune if it nei …

There are some statements, whose consistency (or the consistency of their negation) require the existence of large cardinals (in the sense that if the statement (or its negation) is consistent, then i …

The question is essentially what is asked in the title. I split it into two parts (A) (Arguments supporting the existence of large cardinals) What are the main philosophical arguments in defense o …

For measurable cardinals, the answer is yes and is due to Sy Friedman. See Coding Over a Measurable Cardinal. There is some difficulty to extend the result to the context of Woodin cardinals, see Gen …

Work in $ZF$. Let $j:V\to V$ be a non-trivial elementary embedding which is iterable, so that we can iterate it and form models $M_\alpha, \alpha\in ON,$ with $M_0=V,$ and elementary embeddings $j_{\a …

In his paper Boolean extensions which efface the Mahlo property William Boos proves the following consistency result: Theorem. Assume GCH holds and $\kappa$ is weakly compact. Then there exists a card …

Question 1. Is there a categorical representation of Kunen's inconsistency result? Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and supercomp …

In the paper Proof theory and set theory Takeuti has given such a characterization for measurable cardinals, strongly compact cardinals, supercompact cardinals and even large cardinals. Let me first …

The answer to your question is (almost) yes (almost is because of the addition of DC to the statement). Recently Gabriel Goldberg has proved ''Con(NBG+DC+Reinhardt)$ \implies$ Con(ZFC+I0)''. …

Is the following consistent? $(*)$: There exists a singular cardinal $\kappa$ such that : (1) $\kappa$ is a Jonsson cardinal, (2) $\kappa$ is not a fixed point of the $\aleph-$function, i.e., $\kappa …

It is consistent that $AC$ fails and there exists a non-trivial elementary embedding $j: L(V_{\lambda+1}) \to L(V_{\lambda+1})$ with $crit(j) < \lambda,$ and $\lambda$ has uncountable cofinality. See …

It is well-known that if $AC$ holds and if $j: L(V_{\lambda+1}) \to L(V_{\lambda+1})$ is a non-trivial elementary embedding with $crit(j) < \lambda,$ then $\lambda$ has countable cofinality (and in fa …