Trevor Wilson
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Registered User
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Mar 21 |
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Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal Thanks. By the way, I should probably share my motivation: I think it is interesting that the simultaneous failure of $\square(\kappa)$ and $\square_\kappa$ is strong whereas the failure of either on its own is weak. I have an application where it's not clear that $\neg \square(\kappa) \And \neg \square_\kappa$ is enough, however, so hopefully I can strengthen $\neg \square(\kappa)$ to "$\kappa$ is weakly compact" and strengthen $\neg \square_\kappa$ slightly to something like in the question, but still maintain the property that they are weaker on their own than they are together. |
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Mar 21 |
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Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal ...whereas on the other hand if we don't know that $\kappa$ is weakly compact I don't see any way to get more than two Mahlo cardinals out of it, so I thought maybe it could be forced from two Mahlo cardinals somehow (the first being $\kappa$ and the second becoming $\kappa^+$.) I don't know if this is plausible though. |
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Mar 21 |
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Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal Yes, that is definitely worth mentioning. Ideally, I was hoping for something that implied the stationarity of the set in the question without also implying that $\kappa$ is weakly compact. This is because if we add the additional assumption that $\kappa$ is weakly compact, then $\square(\kappa)$ and $\square_\kappa$ both fail, and the lower bound for this coincides with the current state of the art in inner model theory (I think)... |
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Mar 21 |
asked | Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal |
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Mar 19 |
answered | Relation between $\neg \square(\kappa)$ and the tree property at $\kappa$. |
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Feb 15 |
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Homogeneous Namba-like forcing @alephomega When I use that phrase I usually mean "almost homogeneous forcing", which means that for any two conditions $p$ and $q$ in the forcing poset $\mathbb{P}$ there is an automorphism $\pi$ of $\mathbb{P}$ such that $\pi(p)$ is compatible with $q$. The only consequence of this that I am interested in is that the theory of the forcing extension with parameters from the ground model does not depend on the choice of generic filter. |
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Feb 1 |
revised |
Homogeneous Namba-like forcing clarification |
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Jan 31 |
asked | Homogeneous Namba-like forcing |
