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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 …

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 …

Suppose $\kappa$ is a large cardinal (strong cardinal seems to be enough). Is there a forcing notion $\mathbb{R}$ with the following properties: $(1)$ Forcing with $\mathbb{R}$ adds a club $C$ into $ …

There are several ways to force $GCH$ below $\aleph_\omega$ and $2^{\aleph_\omega}> \aleph_{\omega+1},$ say: 1) The Gitik-Magidor's extender based forcing, see Prikry type Forcings. 2) Woodin's meth …

Consider Prikry's forcing for changing the cofinality of a measurable cardinal into $\omega.$ The forcing has the Prikry property and one can prove this either directly or using Rowbottom's theorem wh …

Does Con(ZFC+there exists a strongly compact cardinal) imply the Con(ZFC+there exists a Shelah cardinal)?

This question is motivated by the work of Ajtai "The complexity of the pigeonhole principle" and similar works. In this paper, the author proves that $PHP_n$, the pigeonhole principle for $n,$ does n …

The following question is motivated by a result of Magidor that it is consistent that the least strongly compact cardinal is the least measurable cardinal. Question. Assume $\kappa$ is a strongly …

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 …

Assume $V=L$ and let $\kappa$ be a Mahlo cardinal. Let $L[G]$ be the generic extension obatined by Mitchell forcing to make $2^{\aleph_0}=\aleph_2=\kappa.$ It is known that in the extension there ar …

Question. Suppose $\kappa$ is a supercompact cardinal and $\lambda > \kappa$ is measurable (or even larger large cardinal if necessary). Is there a set generic extension of the universe in which $\ka …

Prikry forcing can be used to produce a model $V$ of $ZFC$ such that fo rsome cardinal $\kappa$ we have: (1) $\kappa$ is singular in $V$ of cofinality $\omega,$ (2) $\kappa$ is regular (and in fact me …

The first question asks about the global behavior of the power function in the case of finite gaps. Question 1. Fix a natural number $m>1.$ For each limit ordinal $\alpha,$ including $0$, let $\ph …

Is the following equiconsistent with the existence of a strongly compact cardinals: For every $\lambda > \kappa$ there exists a $\lambda$-strongly compact embedding $j: V \to M$ with the additiona …

Given a cardinal $\kappa,$ recall that $X \subset \kappa$ is called fresh (over $V$), if $X \notin V,$ but $X \cap \alpha \in V$ for all $\alpha < \kappa.$ Question. Is it consistent that there exis …