Families of Almost Disjoint Sets

As part of their study of βω — ω and βω1 — ω1, A. Szymanski and H. X. Zhou [3] were able to exploit the following difference between ω, and ω: ω1, contains uncountably many disjoint sets whereas any uncountable family of subsets of ω is, at best, almost disjoint. To translate this distinction between ω1, and ω to a possible distinction between βω1 — ω1, and βω — ω they used the fact that if a pairwise disjoint family of sets and a subset of each member of is chosen then it is trivial to find a single set whose intersection with each member is the chosen set. However, they noticed, it is not clear that the same is true if is only a pairwise almost disjoint family even if we only require equality except on a finite set. But any homeomorphism from βω1 — ω1 to βω — ω would have to carry a disjoint family of subsets of ω1, to an almost disjoint family of subsets of ω with this property. This observation should motivate the following definition.

Download Full-text

Ultrafilters and almost disjoint sets. II

Bulletin of the American Mathematical Society ◽

10.1090/s0002-9904-1975-13715-3 ◽

1975 ◽

Vol 81 (1) ◽

pp. 209-213 ◽

Cited By ~ 4

Author(s):

Karel Prikry

Keyword(s):

Almost Disjoint ◽

Disjoint Sets

Download Full-text

Almost disjoint sets and Martin's axiom

Journal of Symbolic Logic ◽

10.2307/2273124 ◽

1979 ◽

Vol 44 (3) ◽

pp. 313-318 ◽

Cited By ~ 3

Author(s):

Michael L. Wage

Keyword(s):

Partial Order ◽

Martin's Axiom ◽

Martin’S Axiom ◽

Almost Disjoint ◽

Disjoint Sets

AbstractWe present a number of results involving almost disjoint sets and Martin's axiom. Included is an example, due to K. Kunen, of a c.c.c. partial order without property K whose product with every c.c.c. partial order is c.c.c.

Download Full-text

Ordinal definability in Jensen's model

Journal of Symbolic Logic ◽

10.2307/2274192 ◽

1984 ◽

Vol 49 (2) ◽

pp. 608-620 ◽

Cited By ~ 1

Author(s):

Włodzimierz Zadrożny

Keyword(s):

Boolean Algebras ◽

Almost Disjoint ◽

Partial Orderings ◽

Disjoint Sets

In [J] R. Jensen proved that any model of ZFC can be generically extended to a model of ZFC + (∃a ⊆ ω)(V = L[a]). This extension was made via a class P of forcing conditions. One can ask what can be said about the class HOD in such extensions. In particular one can ask whether V = HOD holds in Jensen's model. A full answer to this question is given by the following assertion:Theorem. Let N = L[a] be given by forcing with Jensen's conditions (the classP).Then:1. N ⊨ V ≠ HOD,2. N ⊨ (∀n < ω)(HODn + 1 ⊈ HODω).3. N ⊨ (∀α ≥ ω)(HODα = HODω), or equivalently HODω + 1 = HODω.The result is established by investigating homogeneity properties of certain complete Boolean algebras that are associated with the partial orderings of forcing conditions for coding using almost disjoint sets.We explain shortly the first of our two technical results: It is rather well known that if we have a subset then by performing threefold almost disjoint coding we come to a set a0 ⊆ ω such that ; cf. [JS]. This is achieved by iterating over almost disjoint forcing . That means we first code by a subset , of [ω1ω2) using the almost disjoint forcing conditions . Then we code by aω, contained in [ω, ω1), using forcing with over .

Download Full-text