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Claim. For any $\alpha >0 \colon$ $\mathrm{ZFC} \not \vdash \mathrm{CH}(\aleph_{\alpha})$
Proof. Starting with $L$, we may enlarge the continuum $\beth_1$ arbitrarily without changing cardinals. ...

If $\operatorname{card}(X) > \omega$ is regular, the answer is yes:
Lemma 1. Let $\kappa$ be a regular, uncountable cardinal and let $f \colon \kappa \to \kappa$ be such that $S := \{ \xi < \...

During this year's conferene on inner model theory in Münster, Gabriel Goldberg proved that the so-called Ultrapower Axiom implies that $\mathrm{GCH}$ holds above a supercompact cardinal (and since ...

Note that we can use $g,h$ to code any real of $V$ in a recursive fashion (recursive in the Cohen reals added by $g$ and $h$). In particular we can use them to code $0^{\#}$ -- a real that cannot be ...

Yes, this is consistent.
Consider a (transitive) model of $\mathrm{ZF}$ in which $\omega_1$ has countable cofinality. Fix a strictly increasing, cofinal sequence $(\xi_n \mid n < \omega)$ in $\...

The following is part of unpublished work by Toshimichi Usuba:
Definition. A cardinal $\kappa$ is hyper-huge if for every $\lambda > \kappa$ there is an elementary embedding
$$
j \colon V \to M
$$
...

Simon Thomas already mentioned that a winning strategy can be recursively coded into a real. I thought it might be a good idea to write down one such coding explicitly:
Let $A \subseteq \mathcal{N}$ ...

Fix $(a_n \mid n < \omega)$, $(x_n \mid n < \omega)$ such that $x_n \in E_{a_n}$ for all $n < \omega$. Without loss of generality we may assume
$\{\xi\} \in \{a_n \mid n < \omega \}$ for ...

In my question I already verified that (assuming 1. and 2.) item 3. is provable without assuming that $\pi(r)$ is $k$-solid over $(\mathcal{M},q)$. To see that we can actually drop $k$-solidity in ...

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