Difference between revisions of "PFA"

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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.
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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>
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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}}

Latest revision as of 01:21, 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

  1. 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
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