MAD families and the rationals
نویسنده
چکیده
Rational numbers are used to classify maximal almost disjoint (MAD) families of subsets of the integers. Combinatorial characterization of indestructibility of MAD families by the likes of Cohen, Miller and Sacks forcings are presented. Using these it is shown that Sacks indestructible MAD family exists in ZFC and that b = c implies that there is a Cohen indestructible MAD family. It follows that a Cohen indestructible MAD family is in fact indestructible by Sacks and Miller forcings. A connection with Roitman’s problem of whether d = ω1 implies a = ω1 is also discussed.
منابع مشابه
Very Mad Families
The notion of very mad family is a strengthening of the notion of mad family of functions. Here we show existence of very mad families in different contexts.
متن کاملThe Role of the Scholars` Tradition in Formulating Moral Propositions, with an Emphasis on Muhaqqiq Isfahani`s View
The issue of the source of ethical value is one of the most important issues in the field of metaethics. Moral philosophers have come up with different perspectives on this issue. Some Islamic thinkers, including Muhaqqiqi Isfahani, have considered the rationals` tradition as the formulating cause for moral propositions. This research is intended to emphasize on the viewpoint of Muhaqqiqi Isfah...
متن کاملMad spectra
The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on ω. We treat the problem to characterize those sets A which, in some forcing extension of the universe, can be the mad spectrum. We give a complete solution to this problem under the assumption θ = θ, where θ = min(A). §0. Introduction. Recall that A ⊆ [ω] is called almost disjoint (a. d. for short),...
متن کاملMad families constructed from perfect almost disjoint families
We prove the consistency of b > א1 together with the existence of a Π11definable mad family, answering a question posed by Friedman and Zdomskyy in [7, Question 16]. For the proof we construct a mad family in L which is an א1-union of perfect a.d. sets, such that this union remains mad in the iterated Hechler extension. The construction also leads us to isolate a new cardinal invariant, the Bor...
متن کامل