1,116 research outputs found

    Equivalence of generics

    Full text link
    Given a countable transitive model of set theory and a partial order contained in it, there is a natural countable Borel equivalence relation on generic filters over the model; two are equivalent if they yield the same generic extension. We examine the complexity of this equivalence relation for various partial orders, with particular focus on Cohen and random forcing. We prove, amongst other results, that the former is an increasing union of countably many hyperfinite Borel equivalence relations, while the latter is neither amenable nor treeable.Comment: 18 pages. We have made minor stylistic changes and corrected an error in the statement of Lemma 3.5, now appearing as Lemmas 3.5 and 3.

    Mathias--Prikry and Laver type forcing; Summable ideals, coideals, and ++-selective filters

    Full text link
    We study the Mathias--Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias--Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias--Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of ω\omega-hitting and ω\omega-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.Comment: updated versio

    On a dichotomy related to colourings of definable graphs in generic models

    Full text link
    We prove that in the Solovay model every OD graph G on reals satisfies one and only one of the following two conditions: (I) G admits an OD colouring by ordinals; (II) there exists a continuous homomorphism of G_0 into G, where G_0 is a certain F_sigma locally countable graph which is not R-OD colourable by ordinals in the Solovay model. If the graph G is locally countable or acyclic then (II) can be strengthened by the requirement that the homomorphism is a 1-1 map, i.e. an embedding. As the second main result we prove that Sigma^1_2 graphs admit the dichotomy (I) vs. (II) in set--generic extensions of the constructible universe L (although now (I) and (II) may be in general compatible). In this case (I) can be strengthened to the existence of a Delta^1_3 colouring by countable ordinals provided the graph is locally countable. The proofs are based on a topology generated by \od sets

    Some interesting problems

    Full text link
    This is an update of my problem list

    In Memoriam: James Earl Baumgartner (1943-2011)

    Full text link
    James Earl Baumgartner (March 23, 1943 - December 28, 2011) came of age mathematically during the emergence of forcing as a fundamental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable orders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, he applied his knowledge of set theory to a variety of areas in collaboration with other mathematicians, and he encouraged a community of mathematicians with engaging survey talks, enthusiastic discussions of open problems, and friendly mathematical conversations.Comment: 51 page

    Cardinal invariants of closed graphs

    Full text link
    We study several cardinal characteristics of closed graphs G on compact metrizable spaces. In particular, we address the question when it is consistent for the bounding number to be strictly smaller than the smallest size of a set not covered by countably many compact G-anticliques. We also provide a descriptive set theoretic characterization of the class of analytic graphs with countable coloring number

    Specializing Wide Aronszajn Trees without Adding Reals

    Full text link
    We show that under certain circumstances wide Aronszajn trees can be specialized iteratively without adding reals. We then use this fact to study forcing axioms compatible with CH and list some open problems.Comment: 15 Pages, submitted to the RIMS Set Theory and Infinity 2019 Kokyuroku, second version cleans up the exposition and fixes a gap in the proof of Lemma 3.

    Evasion and prediction --- the Specker phenomenon and Gross spaces

    Full text link
    We study the set--theoretic combinatorics underlying the following two algebraic phenomena. (1) A subgroup G leq Z^omega exhibits the Specker phenomenon iff every homomorphism G to Z maps almost all unit vectors to 0. Let se be the size of the smallest G leq Z^omega exhibiting the Specker phenomenon. (2) Given an uncountably dimensional vector space E equipped with a symmetric bilinear form Phi over an at most countable field KK, (E,Phi) is strongly Gross iff for all countably dimensional U leq E, we have dim(U^perp) leq omega. Blass showed that the Specker phenomenon is closely related to a combinatorial phenomenon he called evading and predicting. We prove several additional results (both theorems of ZFC and independence proofs) about evading and predicting as well as se, and relate a Luzin--style property associated with evading to the existence of strong Gross spaces

    Complete ccc Boolean algebras, the order sequential topology, and a problem of von Neumann

    Full text link
    It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure

    On a Glimm -- Effros dichotomy and an Ulm--type classification in Solovay model

    Full text link
    We prove that in Solovay model every OD equivalence E on reals either admits an OD reduction to the equality on the set of all countable (of length < omega_1) binary sequences, or continuously embeds E_0, the Vitali equivalence. If E is a Sigma_1^1 (resp. Sigma_1^2) relation then the reduction in the ``either'' part can be chosen in the class of all Delta_1 (resp. Delta_2) functions. The proofs are based on a topology generated by OD sets
    • …
    corecore