Some remarks on a question of D. H. Fremlin regarding ε-density

Archive for Mathematical Logic 40 (7):531-540 (2001)
  Copy   BIBTEX

Abstract

We show the relative consistency of ℵ1 satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω1 satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of ω1, and specifically if this and a stronger property hold under MA + ¬CH. Towards this we show that if MA + ¬CH holds, then for every family ? of ℵ1 many infinite subsets of ω1, one can find a family ? of finite subsets of ω1 which is dense in Fremlins sense, and does not contain all finite subsets of any set in ?.We then pose some open problems related to the question

Other Versions

No versions found

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 99,445

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

Trees and Π 1 1 -Subsets of ω1 ω 1.Alan Mekler & Jouko Vaananen - 1993 - Journal of Symbolic Logic 58 (3):1052 - 1070.
Σ1(κ)-definable subsets of H.Philipp Lücke, Ralf Schindler & Philipp Schlicht - 2017 - Journal of Symbolic Logic 82 (3):1106-1131.
The cardinals below | [ ω 1 ] ω 1 |.W. Hugh Woodin - 2006 - Annals of Pure and Applied Logic 140 (1-3):161-232.
Games of length ω1.Itay Neeman - 2007 - Journal of Mathematical Logic 7 (1):83-124.
Cardinal-preserving extensions.Sy D. Friedman - 2003 - Journal of Symbolic Logic 68 (4):1163-1170.
Connected modal logics.Guram Bezhanishvili & David Gabelaia - 2011 - Archive for Mathematical Logic 50 (3-4):287-317.
An infinitary Ramsey property.William J. Mitchell - 1992 - Annals of Pure and Applied Logic 57 (2):151-160.

Analytics

Added to PP
2013-11-23

Downloads
42 (#430,838)

6 months
10 (#288,781)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Džamonja Mirna
University of East Anglia

Citations of this work

Add more citations

References found in this work

[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.
Some results on consecutive large cardinals.Arthur W. Apter - 1983 - Annals of Pure and Applied Logic 25 (1):1-17.
A Relativization of Axioms of Strong Infinity to ^|^omega;1.Gaisi Takeuti - 1970 - Annals of the Japan Association for Philosophy of Science 3 (5):191-204.

Add more references