Definable filters and N*

A talk presenting joint work with Will Brian where we construct maximal and minimal idempotent ultrafilters.

The talk was presented at the "Sets and Computations" workshop held at the Institute of Mathematics of the National University of Singapore.

Transcript

1. Definable semifilters and N∗ (joint work with Will Brian) Department of Logic FF UK Jonathan Verner
2. Semifilters Department of Logic FF UK Definition: A semifilter on ω is a family of infinite subsets of ω closed upwards with respect to ⊆
3. Semifilters Department of Logic FF UK Definition: A semifilter on ω is a family of infinite subsets of ω closed upwards with respect to ⊆and closed under finite modifications.
4. Semifilters Department of Logic FF UK Definition: A semifilter on ω is a family of infinite subsets of ω closed upwards with respect to ⊆and closed under finite modifications. Examples: filters
5. Semifilters Department of Logic FF UK Definition: A semifilter on ω is a family of infinite subsets of ω closed upwards with respect to ⊆and closed under finite modifications. Examples: filters, sets positive with respect to a filter (also called co-ideals, superfilters or ramsey semifilters)
6. Semifilters Department of Logic FF UK Definition: A semifilter on ω is a family of infinite subsets of ω closed upwards with respect to ⊆and closed under finite modifications. Examples: filters, sets positive with respect to a filter (also called co-ideals, superfilters or ramsey semifilters), sets containing arbitrarily long intervals (thick sets). ▶ A very general notion, not really interesting on its own, just a shortcut.
7. Semifilters Department of Logic FF UK Definition: A semifilter on ω is a family of infinite subsets of ω closed upwards with respect to ⊆and closed under finite modifications. Examples: filters, sets positive with respect to a filter (also called co-ideals, superfilters or ramsey semifilters), sets containing arbitrarily long intervals (thick sets). ▶ A very general notion, not really interesting on its own, just a shortcut. ▶ There is a book on them ([?])
8. Semifilters Department of Logic FF UK Definition: A semifilter on ω is a family of infinite subsets of ω closed upwards with respect to ⊆and closed under finite modifications. Examples: filters, sets positive with respect to a filter (also called co-ideals, superfilters or ramsey semifilters), sets containing arbitrarily long intervals (thick sets). ▶ A very general notion, not really interesting on its own, just a shortcut. ▶ There is a book on them ([?])
9. Definable Semifilters Department of Logic FF UK ▶ Characterization of Fσ -semifilters ▶ Baire semifilters are meager or co-meager ▶ meager/co-meager are RB-above Fr/Fin+
10. p,t Department of Logic FF UK ▶ Semifilters are just partial orders, can define p, t ▶ Rudin-Blass reductions ▶ Gδ → p = t ▶ Fσ, Co − meager & p = t = ℵ0 ▶ Does p always equal t? ▶ Consistently p < pS
11. Ultrafilters on S Department of Logic FF UK ▶ consider S as a partial order; ultrafilters = maximal filters ▶ d = c gives P-ultrafilters on every Gδ semifilter ▶ weak P-ultrafilters on every co-meager semifilter in ZFC
12. Applications Department of Logic FF UK Idempotents in βN ▶ (βN, +) is a topological semigroup ▶ minimal left ideals, idempotents, ordering of idempotents ▶ Hindman, Strauss, 9.25&9.26: Are there minimal&maximal idempotents? ▶ I learned last week that Zelenyuk had, independently, answered the question
13. Applications Department of Logic FF UK Idempotents in βN ▶ Thick sets ▶ maximal filters on thick sets = minimal left ideals ▶ weak P-sets = minimal&maximal idempotents
14. Applications Department of Logic FF UK Interesting ultrafilters Theorem: An ultrafilter on a co-ideal is a (full) ultrafilter. We get (weak) P-points consisting of interesting sets ▶ van der Waerden sets ▶ non summable sets ▶ sets of edges containing arbitrarily large complete subgraphs Some of these ultrafilters cannot be selective