Thin Stationary Sets and Disjoint Club Sequences
نویسنده
چکیده
We describe two opposing combinatorial properties related to adding clubs to ω2: the existence of a thin stationary subset of Pω1 (ω2) and the existence of a disjoint club sequence on ω2. A special Aronszajn tree on ω2 implies there exists a thin stationary set. If there exists a disjoint club sequence, then there is no thin stationary set, and moreover there is a fat stationary subset of ω2 which cannot acquire a club subset by any forcing poset which preserves ω1 and ω2. We prove that the existence of a disjoint club sequence follows from Martin’s Maximum and is equiconsistent with a
منابع مشابه
Some applications of mixed support iterations
We give some applications of mixed support forcing iterations to the topics of disjoint stationary sequences and internally approachable sets. In the first half of the paper we study the combinatorial content of the idea of a disjoint stationary sequence, including its relation to adding clubs by forcing, the approachability ideal, canonical structure, the proper forcing axiom, and properties r...
متن کاملClub and Stationary Sets in ω1, Probability Measures, and Ulam Matrices
In this talk I will discuss some interesting subsets of ω1 – club-sets, Borel subsets, stationary and bi-stationary subsets of ω1 – and the theorem about them that has been most useful to me over the years, called Fodor’s Theorem. The question “How many types of stationary sets exist in ω1?” will be a recurring theme. (Spoiler alert: there are uncountably many really distinct stationary subsets...
متن کاملToward the Consistency Strength of Stationary Set Reflection on Small Cardinals
OF THE DISSERTATION Toward the Consistency Strength of Stationary Set Reflection on Small Cardinals By Cynthia Northrup Doctor of Philosophy in Mathematics University of California, Irvine, 2015 Professor Dr. Martin Zeman, Chair A set S ⊆ ω3 is stationary in ω3 if it intersects all closed and unbounded subsets in ω3. We say that S reflects below ω3 if there is some α < ω3 for which S ∩ α is sta...
متن کاملClub-guessing, stationary reflection, and coloring theorems
We obtain strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. Along the way, we establish new results in club-guessing and in the general theory of ideals.
متن کاملStrongly adequate sets and adding a club with finite conditions
We continue the study of adequate sets which we began in [2] by introducing the idea of a strongly adequate set, which has an additional requirement on the overlap of two models past their comparison point. We present a forcing poset for adding a club to a fat stationary subset of ω2 with finite conditions, thereby showing that a version of the forcing posets of Friedman [1] and Mitchell [3] fo...
متن کامل