Difference between revisions of "PFA"
From Cantor's Attic
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A cardinal $\kappa$ is a ''PFA cardinal'' if $\kappa$ is not zero and the canonical forcing of the PFA of length $\kappa$, which is the countable support iteration that at each stage $\gamma$ forces with the lottery sum of all minimal-rank proper partial orders $\mathbb{Q}$ for which there is a family $\cal{D}$ of $\omega_1$ many dense sets in $\mathbb{Q}$ for which there is no filter in $\mathbb{Q}$ meeting them, forces PFA. | A cardinal $\kappa$ is a ''PFA cardinal'' if $\kappa$ is not zero and the canonical forcing of the PFA of length $\kappa$, which is the countable support iteration that at each stage $\gamma$ forces with the lottery sum of all minimal-rank proper partial orders $\mathbb{Q}$ for which there is a family $\cal{D}$ of $\omega_1$ many dense sets in $\mathbb{Q}$ for which there is no filter in $\mathbb{Q}$ meeting them, forces PFA. | ||
− | Every supercompact cardinal is a PFA cardinal. It is not yet clear whether the converse is true. | + | Every [[supercompact]] cardinal is a PFA cardinal. It is not yet clear whether the converse is true. |
− | Viale and Weiß have shown that an inaccessible PFA cardinal $\kappa$ for which the canonical forcing of the PFA makes $\kappa$ become $\omega_2$ is supercompact.<cite> | + | Viale and Weiß have shown that an [[inaccessible]] PFA cardinal $\kappa$ for which the canonical forcing of the PFA makes $\kappa$ become $\omega_2$ is supercompact.<cite> |
VialeWeiss2011:OnConsistencyStrengthPFA</cite> | VialeWeiss2011:OnConsistencyStrengthPFA</cite> | ||
{{References}} | {{References}} |
Revision as of 00:21, 10 October 2017
A cardinal $\kappa$ is a PFA cardinal if $\kappa$ is not zero and the canonical forcing of the PFA of length $\kappa$, which is the countable support iteration that at each stage $\gamma$ forces with the lottery sum of all minimal-rank proper partial orders $\mathbb{Q}$ for which there is a family $\cal{D}$ of $\omega_1$ many dense sets in $\mathbb{Q}$ for which there is no filter in $\mathbb{Q}$ meeting them, forces PFA.
Every supercompact cardinal is a PFA cardinal. It is not yet clear whether the converse is true.
Viale and Weiß have shown that an inaccessible PFA cardinal $\kappa$ for which the canonical forcing of the PFA makes $\kappa$ become $\omega_2$ is supercompact.[1]
References
- Viale, Matteo and Weiß, Christoph. On the consistency strength of the proper forcing axiom. Advances in Mathematics 228(5):2672--2687, 2011. arχiv MR bibtex