Difference between revisions of "PFA"

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(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 *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.  
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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.  
  
 
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.