1,116 research outputs found
Equivalence of generics
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
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 -hitting and
-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
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
In Memoriam: James Earl Baumgartner (1943-2011)
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
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
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
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
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
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
- …