The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters

Mathematical Logic Quarterly 59 (4-5):258-267 (2013)
  Copy   BIBTEX

Abstract

Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, (ℵ) iff (2ℵ). iff “ is a continuous image of ” iff “ has a free open ultrafilter ” iff “every countably infinite subset of has a limit point”. implies “every open filter on extends to an open ultrafilter” implies “has an open ultrafilter” implies It is relatively consistent with that (ω) holds, whereas (ω) fails. In particular, none of the statements given in (2) implies (ω).

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 94,549

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Saturating ultrafilters on N.D. H. Fremlin & P. J. Nyikos - 1989 - Journal of Symbolic Logic 54 (3):708-718.
Selective and Ramsey Ultrafilters on G-spaces.Oleksandr Petrenko & Igor Protasov - 2017 - Notre Dame Journal of Formal Logic 58 (3):453-459.
The axiom of choice holds iff maximal closed filters exist.Horst Herrlich - 2003 - Mathematical Logic Quarterly 49 (3):323.
The Ultrafilter Closure in ZF.Gonçalo Gutierres - 2010 - Mathematical Logic Quarterly 56 (3):331-336.
Stable ordered union ultrafilters and cov.David José Fernández-bretón - 2019 - Journal of Symbolic Logic 84 (3):1176-1193.
Ultrafilters on ω.James E. Baumgartner - 1995 - Journal of Symbolic Logic 60 (2):624-639.
Logics From Ultrafilters.Daniele Mundici - forthcoming - Review of Symbolic Logic:1-18.

Analytics

Added to PP
2013-12-01

Downloads
20 (#776,057)

6 months
10 (#397,263)

Historical graph of downloads
How can I increase my downloads?

References found in this work

The Axiom of Choice.Thomas J. Jech - 1973 - Amsterdam, Netherlands: North-Holland.
On generic extensions without the axiom of choice.G. P. Monro - 1983 - Journal of Symbolic Logic 48 (1):39-52.

Add more references