Stefan Mesken
• Member for 7 years, 3 months
• Last seen more than 1 year ago

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

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If $\operatorname{card}(X) &gt; \omega$ 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 $S := \{ \xi &lt; \... View answer 2 answers 18 votes 1k views 7 votes During this year's conferene on inner model theory in Münster, Gabriel Goldberg proved that the so-called Ultrapower Axiom implies that$\mathrm{GCH}$holds above a supercompact cardinal (and since ... View answer 2 answers 11 votes 331 views Accepted answer 6 votes Note that we can use$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 ... View answer 1 answers 3 votes 106 views Accepted answer 5 votes Yes, this is consistent. Consider a (transitive) model of$\mathrm{ZF}$in which$\omega_1$has countable cofinality. Fix a strictly increasing, cofinal sequence$(\xi_n \mid n &lt; \omega)$in$\...

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The following is part of unpublished work by Toshimichi Usuba: Definition.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

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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 $A \subseteq \mathcal{N}$ ...

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