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By Kanamori, ``The higher infinite'', Theorem 21.1 (due to Kunen), the existence of $0^\sharp$ is equivalent to the existence of an iterable $L-$ultrafilter. Also note that if $j: L \to L$ is a non-t …

As it is stated by Yair Hayut, in the comment above, statement $(1)$ implies statement $(2)$. Let's show that statement $(2)$ does not imply statement $(1)$. Let $u(\kappa, \lambda)=cf(P_\kappa(\lam …

As you are mainly interested in Shelah cardinals, let me add a few more remarks about them. Given a Shelah cardinal $\kappa$, let $wt(\kappa),$ the witnessing number of $\kappa,$ be the least $\lambda …

If $\kappa$ is extendible, then there exists a proper class of inaccessible cardinals (and even more).

Let me add one extra example that might be interesting. Let $\pi_n^m$ and $\sigma_n^m$ denote respectively the least $\Pi_n^m$-indescribable and the least $\Sigma_n^m$-indescribable cardinal (if th …

I don't know to what extend the following answers your question, but a surprising result of Woodin says that the extension of the Inner Model Program to the level of one supercompact cardinal must yi …

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, which part of the argu …

I may note that a cardinal of type $A$ may has more consistency strength of a cardinal of type $B$, while the smallest cardinal of type $A$ is smaller than the least cardinal of type $B$ (assuming car …

Maybe a comment: In the paper ``A large cardinal in the constructible universe'' Silver shows that if $\kappa\to (\alpha)^{<\aleph_0}$ for all countable $\alpha,$ then the same is true for $\kappa$ i …

The answer to your question is yes. During the "IPM conference on set theory and model theory" James cummings gave me the basic idea of the proof of the following theorem: Theorem. Assuming the exist …

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 …

Let us call a cardinal $\kappa, \kappa-$compact, if every $\kappa-$complete filter on $\kappa$ extends to a $\kappa-$complete ultrafilter. The following is proved in Gitik's paper On measurable cardi …

The following is due to Woodin: Theorem. Assume $ZF$+ there exists a Reinhardt cardinal + there exists a proper class of supercompact cardinals is consistent. Then there exists a genric extension of …

Let me add more examples: If we consider the global behavior of the power function, then we have for example: (A) (Foreman-Woodin): $F$ can be such that $F(\alpha)>\alpha+\omega,$ all $\alpha$ ( …

The following results are obtained in a joint work with Yair Hayut and are now presented in our joint paper On Foreman's maximality principle: Theorem 1. (Assuming the existence of a strong cardin …