2015/10/19 by Farkas, Barnabás, Khomskii, Yurii, Vidnyánszky, Zoltán · 1 citation
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1510.05699
We investigate families of subsets of ω with almost disjoint refinements in the classical case as well as with respect to given ideals on ω. More precisely, we study the following topics and questions: 1) Examples of projective ideals. 2) We prove the following generalization of a result due to J. Brendle: If V⊆ W are transitive models, ω1W⊆ V, P(ω)∩ V\not = P(ω)∩ W, and I is an analytic or coanalytic ideal coded in V, then there is an I-almost disjoint refinement (I-ADR) of I+∩ V in W, that is, a family \AX:X\inI+∩ V\∈ W such that (i) AX⊆ X, AX∈ I+ for every X and (ii) AX∩ AY\inI for every distinct X and Y. 3) The existence of perfect I-almost disjoint (I-AD) families, and the existence of a "nice" ideal I on ω with the property: Every I-AD family is countable but I is nowhere maximal. 4) The existence of (I,Fin)-almost disjoint refinements of families of I-positive sets in the case of everywhere meager (e.g. analytic or coanalytic) ideals. We prove a positive result under Martin's Axiom. 5) Connections between classical properties of forcing notions and adding mixing reals (and mixing injections), that is, a (one-to-one) function f:ω→ω such that |f[X]∩ Y|=ω for every X,Y∈ [ω]ω∩ V.