The next best thing to a Ppoint
Abstract
We study ultrafilters on $\omega^2$ produced by forcing with the quotient of $\scr P(\omega^2)$ by the Fubini square of the Fréchet filter on $\omega$. We show that such an ultrafilter is a weak Ppoint but not a Ppoint and that the only nonprincipal ultrafilters strictly below it in the RudinKeisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest squarebracket partition relations that are possible for a nonPpoint. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukeyabove $[\omega_1]^{<\omega}$, and it has only continuum many ultrafilters Tukeybelow it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters.
 Publication:

arXiv eprints
 Pub Date:
 August 2013
 arXiv:
 arXiv:1308.3790
 Bibcode:
 2013arXiv1308.3790B
 Keywords:

 Mathematics  Logic
 EPrint:
 Submitted