# Difference between revisions of "PFA"

From Cantor's Attic

(Created page with "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 ...") |
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− | A cardinal $\kappa$ is a | + | 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. |

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. |

## Revision as of 18:21, 24 March 2014

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.

Every supercompact cardinal is a PFA cardinal. It is not yet clear whether the converse is true.