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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 …

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 …

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 …

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 …

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 …

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 …

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

The concept is inconsistent. Consider $\mathbb{P}=Add(\omega, \kappa)_L=Add(\omega, \kappa),$ where $\kappa$ is $(2^{\aleph_0})^+$ of $V$. Forcing with $\mathbb{P}$ over $V$ doesn't collapse cardinal …

Suppose $\kappa$ is a singular cardinal and there are $cf(\kappa)$-many measurable cardinals $\lambda < \kappa$ with $o(\lambda)=\lambda^{++}$ cofinal in $\kappa.$ Then you can perform a Prikry type i …

If you start with a strong (or a supercompact cardinal) and if you force with Radin forcing $\mathbb{R}_u$, for some suitable $u$, then you can preserve the full strength (or supercompactness) of $\ka …

It follows from the work of Gitik and Mitchell Indiscernible sequences for extenders, and the singular cardinal hypothesis that the hypothesis implies the existence of an inner model with overlapping …

Today I saw the following paper in which probabilistic arguments are used in a forcing argument: Halfway New Cardinal Characteristics. See the proof of 3.4. The paper is written by Jörg Brendle, Lor …

Here is an answer to question 2. First note that it suffices to consider the case where $\lambda=\kappa.$ For regular $\kappa,$ you may consider the following graph: The point is that given less t …

Suppose $GCH$ holds, $U$ is a normal measure on $\kappa$ and $j: V \to M$ is an ultrapower embedding. Let $F$ be an Easton function on regular cardinals and $P_F$ be the corresponding Easton forcing …

When dealing with the singular cardinals hypothesis ($SCH$), one may face with many such examples, let me say a few: $\star_1:$ The consistency of the failure of $SCH$ was proved first by silver usin …