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TOWERS IN FILTERS, CARDINAL INVARIANTS, AND LUZIN TYPE FAMILIES
| Content Provider | Scilit |
|---|---|
| Author | Brendle, Jörg Farkas, Barnabás Verner, Jonathan |
| Copyright Year | 2018 |
| Description | We investigate which filters onωcan contain towers, that is, a modulo finite descending sequence without any pseudointersection (in ${[\omega ]^\omega }$ ). We prove the following results: (1)Many classical examples of nice tall filters contain no towers (in ZFC). (2)It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well). (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)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 $F \in {\cal F}$ . |
| Related Links | http://arxiv.org/pdf/1605.04735 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/3EDA30734A6C3E05ECC495A6AA278BDF/S0022481217000524a.pdf/div-class-title-towers-in-filters-cardinal-invariants-and-luzin-type-families-div.pdf |
| Ending Page | 1062 |
| Page Count | 50 |
| Starting Page | 1013 |
| ISSN | 00224812 |
| e-ISSN | 19435886 |
| DOI | 10.1017/jsl.2017.52 |
| Journal | The Journal of Symbolic Logic |
| Issue Number | 3 |
| Volume Number | 83 |
| Language | English |
| Publisher | Cambridge University Press (CUP) |
| Publisher Date | 2018-09-01 |
| Access Restriction | Open |
| Subject Keyword | The Journal of Symbolic Logic Borel Ideal Analytic Ideal Summable Ideal Density Zero Ideal Mathias–prikry Forcing Laver–prikry Forcing Cardinal Invariants of the Continuum Katĕtov Order blass Order Luzin Type Family Near Coherence of Filters |
| Content Type | Text |
| Resource Type | Article |
| Subject | Philosophy Logic |
