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Stefan Mesken's user avatar
Stefan Mesken's user avatar
Stefan Mesken
  • Member for 10 years, 4 months
  • Last seen more than 4 years ago
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What is the error in this disproof of the $\Omega$-conjecture?
I think I am mainly confused by the vagueness of OP's post in several instances. Maybe it is possible to fill in some of those gaps? E.g. how and where exactly do we compute $\Sigma_2$ truths and how would that lead to a truth predicate for $V$?
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What is the error in this disproof of the $\Omega$-conjecture?
But you evaluate the truth of $\Sigma_2$ statements in $H(\delta_0^+)$ and in there, $\delta_0$ is still Woodin.
awarded
 Citizen Patrol
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What is the error in this disproof of the $\Omega$-conjecture?
@Joel Yes, right after posting my comment, I remembered this result by Usuba.
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Cardinality of the set of functions commuting with $f:X\to X$
I truly like the geometric picture this answer provides.
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Cardinality of the set of functions commuting with $f:X\to X$
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Cardinality of the set of functions commuting with $f:X\to X$
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Cardinality of the set of functions commuting with $f:X\to X$
Also note that if $\operatorname{cf}(\kappa) > \omega$, the approach in Lemma 1 still yields $\operatorname{cf}(\kappa)$ many functions commuting with $f$.
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Cardinality of the set of functions commuting with $f:X\to X$
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Cardinality of the set of functions commuting with $f:X\to X$
@Pietro To be honest, I didn't read the proof I linked. My claimed proof was purely based on Dominic's comment below that answer. So, until I have more time to investigate, exclude the case $\kappa = \omega$ from my answer. I'll handle this case asap.
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Cardinality of the set of functions commuting with $f:X\to X$
I think I can prove that we have at least $\operatorname{cf}(\kappa)$ many functions that commutate with any given $f \colon \kappa \to \kappa$. But unfortunately I have to run some other errands first.
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Cardinality of the set of functions commuting with $f:X\to X$
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Commuting functions and fixpoints
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Approve
Commuting functions and fixpoints
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Equivalent definitions of Woodin cardinals in $\operatorname{ZFC}_{-}/\operatorname{ZFC}^{-}$
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Equivalent definitions of Woodin cardinals in $\operatorname{ZFC}_{-}/\operatorname{ZFC}^{-}$
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Equivalent definitions of Woodin cardinals in $\operatorname{ZFC}_{-}/\operatorname{ZFC}^{-}$
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Equivalent definitions of Woodin cardinals in $\operatorname{ZFC}_{-}/\operatorname{ZFC}^{-}$
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What is a 'power admissible model'?
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What is a 'power admissible model'?
Right. On first instinct I thought that adding the predicate $\mathcal{P}$ may end up pushing the complexity of the powerset axiom to $\Delta_0^{\mathcal{P}}$, but that's not the case. After checking the papers once again, I'm fairly convinced that Steel's power admissibility is the one you linked to, but without the powerset axiom. Thanks for your help!
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