Stefan Mesken
  • Member for 7 years, 3 months
  • Last seen more than 1 year ago
1 answers
9 votes
782 views
Is it known whether or not $\aleph_\alpha=\beth_\alpha$ can be proven by ZFC?
Accepted answer
8 votes

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. ...

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4 answers
13 votes
675 views
Cardinality of the set of functions commuting with $f:X\to X$
8 votes

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 < \...

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2 answers
18 votes
1k views
Does $V = \textit{Ultimate }L$ imply GCH?
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 ...

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2 answers
11 votes
331 views
Does $0^{\#}$ imply the failure of upward directedness in the set generic universe over $L$?
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 ...

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1 answers
3 votes
106 views
Function $f:\kappa\to\alpha$ with small fibers where $\alpha\in\kappa$
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 $(\xi_n \mid n < \omega)$ in $\...

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4 answers
22 votes
2k views
Anti-large cardinal principles
5 votes

The following is part of unpublished work by Toshimichi Usuba: Definition.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 $$ j \colon V \to M $$ ...

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1 answers
1 votes
322 views
Defining cones and Turing cones
Accepted answer
4 votes

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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1 answers
6 votes
338 views
Regarding extenders
3 votes

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 ...

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1 answers
5 votes
219 views
Mitchell, Steel. FSIT. Lemma 2.8: Is $k$-solidity actually needed?
Accepted answer
3 votes

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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