# Tagged Questions

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### Inaccessible becomes successor of singular

Is it possible, starting from any large cardinal assumption, to find a countably closed forcing $\mathbb{P}$ such that for some inaccessible $\kappa$, $\Vdash_\mathbb{P} "\kappa = \lambda^+$ and ...
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### Strongly compact cardinal with bad covering properties

This is a continuation of the question covering properties of strongly compact embedding. Recall that a cardinal $\kappa$ is $\nu$-strongly compact cardinal if there is an elementary embedding ...
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### Namba forcing and semiproperness

This question is the result of leaving "Proper and Improper Forcing" on my nightstand by accident. Is the statement "Namba forcing is semiproper" known to be equiconsistent with some more standard ...
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### Can a Measureable Cardinal Become the Least Weakly Compact Cardinal in a Forcing Extension?

I am trying to establish whether it is consistent that some property holds at the least weakly compact cardinal. I know that the property holds at measureables. Hence (hoping everything else goes ...
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### A proposed axiom of Laver (updated)

A few months back, I posted a question asking about a proposed axiom of Laver's and I, unfortunately, left out a critical piece. Here is the full axiom: (L) Some elementary embedding ...
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### PFA and the compactness/incompactness of $\omega_2$

There are many consequences of the Proper Forcing Axiom (PFA) which essentially say that $\omega_2$ has a lot of compactness. For instance, Matteo Viale and Christoph Weiss have a few papers in ...
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### Why is this theorem (about $L(P(\omega_1))^V$ and $L(P(\omega_1))^{V[G]}$) nice?

I was recently told that the following (due to M. Viale) is a nice theorem: Suppose there are arbitrarily large supercompacts, and $\mathrm{MM}$ holds in $V$. Let $G$ be generic for a proper ...
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### Adding large sets by countable conditions preserving the GCH

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 ...
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### $< \aleph_1-$support Product of Cohen forcings

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 ...
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### Questions about $\aleph_1-$closed forcing notions

"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 ...
This question is related to this one. The setup is as follows: In $V$, $\kappa$ is supercompact and $\mathbb{P} = \mathrm{Coll}(\kappa, \aleph_2)$ is the Levy collapse. $G$ is $(V,P)$-generic, and ...
### failure of $\square(\kappa)$ at an inaccessible $\kappa$
How can we force the failure of $\square(\kappa)$ at an inaccessible $\kappa$, where $\square(\kappa)$ is defined as follows: There is a sequence $(C_i:i< \kappa)$ such that: (1) \$C_{i+1} = ...