Fabiana Castiblanco: Capturing tree forcing notions and some preservation results

The seminar meets on Wednesday October 24th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Fabiana Castiblanco — Capturing tree forcing notions and some
preservation results

In this talk, we will introduce the concept of capturing forcing notions
in order to show that various tree posets such as Sacks (S), Silver (V),
Mathias (M), Laver (L) and Miller (ML) forcing preserve the existence of
sharps for reals. Furthermore, these tree forcing notions preserve
levels of Projective Determinacy. As a consequence of this fact we
obtain that Σ^1_{n+3}-P-absoluteness holds for P∈T := {S, V, M, L, ML}
under the assumption of Π^1_{n+1}-determinacy.
If time permits, as an application of our results, we will see that if
Π^1_{n+1}-determinacy holds, each P∈T does not add new orbits to
∆^1_{n+3}-thin transitive relations.


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