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TOWERS IN FILTERS, CARDINAL INVARIANTS, AND LUZIN TYPE FAMILIES

Published online by Cambridge University Press:  23 October 2018

JÖRG BRENDLE
Affiliation:
GRADUATE SCHOOL OF SYSTEM INFORMATICS KOBE UNIVERSITY KOBE, JAPANE-mail:brendle@kurt.scitec.kobe-u.ac.jp
BARNABÁS FARKAS
Affiliation:
KURT GÖDEL RESEARCH CENTER UNIVERSITY OF VIENNA VIENNA, AUSTRIAE-mail:barnabasfarkas@gmail.com
JONATHAN VERNER
Affiliation:
DEPARTMENT OF LOGIC CHARLES UNIVERSITY PRAGUE, CZECH REPUBLICE-mail:jonathan.verner@matfyz.cz

Abstract

We investigate which filters on ω can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in ${[\omega ]^\omega }$). We prove the following results:

  1. (1) Many classical examples of nice tall filters contain no towers (in ZFC).

  2. (2) It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well).

  3. (3) It is consistent that all towers generate nonmeager filters (this answers a question of P. Borodulin-Nadzieja and D. Chodounský), in particular (consistently) Borel filters do not contain towers.

  4. (4) The statement “Every ultrafilter contains towers.” is independent of ZFC (this improves an older result of K. Kunen, J. van Mill, and C. F. Mills).

Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters (${\rm{ad}}{{\rm{d}}^{\rm{*}}}\left( {\cal F} \right)$, ${\rm{co}}{{\rm{f}}^{\rm{*}}}\left( {\cal F} \right)$, ${\rm{no}}{{\rm{n}}^{\rm{*}}}\left( {\cal F} \right)$, and ${\rm{co}}{{\rm{v}}^{\rm{*}}}\left( {\cal F} \right)$), and the existence of Luzin type families (of size $\ge {\omega _2}$), that is, if ${\cal F}$ is a filter then ${\cal X} \subseteq {[\omega ]^\omega }$ is an ${\cal F}$-Luzin family if $\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$ is countable for every $F \in {\cal F}$.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2018 

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