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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 …

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 $\kappa$ be an inaccessible cardinal. Is there any forcing notion $P$ with the following properties: 1-$P$ preserves GCH and the strong inaccessibility of $\kappa$, 2-$P$ adds a subset of $\kapp …

A well-known result of Kunen says that there is no non-trivial elementary embedding $j: V \rightarrow V.$ There are several proofs of this theorem (see Kanamori, The higher infinite). I wonder to know …

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 …

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

1-Let $P\in L$ be a nontrivial set forcing or even a tame class forcing (tameness in the sense of Sy Friedman; see for example his Handbook paper), and let $G$ be $P-$generic over $L$. Then it is well …

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 …

Suppose $\kappa$ is an inaccessible cardinal, and let $P$ be the $< \aleph_1-$support product of $Add(\alpha^{++}, 1)$ for singular cardinals $\alpha < \kappa.$ 1- Does this forcing preserve cardina …

For an infinite cardinal $\kappa$ and an ordinal $\lambda>\kappa,$ $\kappa$ is called $\lambda-$strong, if there is a non-trivial elementary embedding $j: V \rightarrow M$ with $crit(j)=\kappa$ such t …

An unpublished result of Woodin says the following: Theorem. Assuming the existence of large cardinals, it is consistent that $\forall \lambda, 2^{\lambda}=\lambda^{++}.$ In the paper "The generaliz …

There are many $ZFC$ results that their proof uses forcing: The idea is that we force the statement to be true, and then using absoluteness (or other reasons) we conclude that the result is true in $Z …

"Foreman`s maximality principle" states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including: 1) $GCH$ fails everywhere …

1-Let $P=Add(\omega_1, \kappa)$, and let $D$ be a dense open subset of $P$. Then there is a dense subset $S$ of $D$ such that for every $f \in S$ and any $g \in P$, if $domf=domg$ and the set $\{ \bet …

The question is motivated by Toni's question "Approximation of infinite set in generic extension" (see Approximation of infinite set in generic extension). Before I state the question, let me add som …