Ultrafilters without p-point quotients
Duration: 55 mins 54 secs
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Description: |
Goldstern, M (Technische Universität Wien)
Friday 28 August 2015, 10:00-11:00 |
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Created: | 2015-09-02 17:58 |
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Collection: | Mathematical, Foundational and Computational Aspects of the Higher Infinite |
Publisher: | Isaac Newton Institute |
Copyright: | Goldstern, M |
Language: | eng (English) |
Abstract: | A p-point is a nonprincipal ultrafilter on the set N of natural numbers which has the property that for every countable family of filter sets there is a pseudointersection in the filter, i.e. a filter set which is almost contained in each set of the family. Equivalently, a p-point is an element of the Stone-Cech remainder beta(N) minus N whose neighborhood filter is closed under countable intersections.
It is well known that p-points "survive" various forcing iterations, that is: extending a universe V with certain forcing iterations P will result in a universe V' in which all (or at least: certain well-chosen) p-points are still ultrafilter bases in the extension. This shows that the sentence "The continuum hypothesis is false, yet there are aleph1-generated ultrafilters, namely: certain p-points" is relatively consistent with ZFC. In a joint paper with Diego Mejia and Saharon Shelah (still in progress) we construct ultrafilters on N which are, on the one hand, far away from being p-points (there is no Rudin-Keisler quotient which is a p-point), but on the other hand can survive certain forcing iterations adding reals but killing p-points. This shows that non-CH is consistent with small ultrafilter bases AND the nonexistence of p-points. |
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