100 research outputs found
Total Representations
Almost all representations considered in computable analysis are partial. We
provide arguments in favor of total representations (by elements of the Baire
space). Total representations make the well known analogy between numberings
and representations closer, unify some terminology, simplify some technical
details, suggest interesting open questions and new invariants of topological
spaces relevant to computable analysis.Comment: 30 page
Techniques for approaching the dual Ramsey property in the projective hierarchy
We define the dualizations of objects and concepts which are essential for
investigating the Ramsey property in the first levels of the projective
hierarchy, prove a forcing equivalence theorem for dual Mathias forcing and
dual Laver forcing, and show that the Harrington-Kechris techniques for proving
the Ramsey property from determinacy work in the dualized case as well
Uniformity, Universality, and Computability Theory
We prove a number of results motivated by global questions of uniformity in
computability theory, and universality of countable Borel equivalence
relations. Our main technical tool is a game for constructing functions on free
products of countable groups.
We begin by investigating the notion of uniform universality, first proposed
by Montalb\'an, Reimann and Slaman. This notion is a strengthened form of a
countable Borel equivalence relation being universal, which we conjecture is
equivalent to the usual notion. With this additional uniformity hypothesis, we
can answer many questions concerning how countable groups, probability
measures, the subset relation, and increasing unions interact with
universality. For many natural classes of countable Borel equivalence
relations, we can also classify exactly which are uniformly universal.
We also show the existence of refinements of Martin's ultrafilter on Turing
invariant Borel sets to the invariant Borel sets of equivalence relations that
are much finer than Turing equivalence. For example, we construct such an
ultrafilter for the orbit equivalence relation of the shift action of the free
group on countably many generators. These ultrafilters imply a number of
structural properties for these equivalence relations.Comment: 61 Page
The Theory of Countable Analytical Sets
The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n a largest countable â_n^1 set of reals, C_n (this is also true for n even, replacing â_n^1 by ÎŁ_n^1 and has been established earlier by Solovay for n = 2 and by Moschovakis and the author for all even n > 2). The internal structure of the sets C_n is then investigated in detail, the point of departure being the fact that each C_n is a set of Î_n^1-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, Ï-models of analysis, higher-level analogs of the constructible universe, inductive definability, etc
Forcing the -Reduction Property and a Failure of -Uniformization
We force over the constructible universe to obtain a model of the
-reduction property, thus lowering the best known large cardinal
strength from the existence of to just ZFC. In this model the
-uniformization property fails, which separates these two principles
for the first time.Comment: 27 page
Automorphism groups of randomized structures
We study automorphism groups of randomizations of separable structures, with
focus on the -categorical case. We give a description of the
automorphism group of the Borel randomization in terms of the group of the
original structure. In the -categorical context, this provides a new
source of Roelcke precompact Polish groups, and we describe the associated
Roelcke compactifications. This allows us also to recover and generalize
preservation results of stable and NIP formulas previously established in the
literature, via a Banach-theoretic translation. Finally, we study and classify
the separable models of the theory of beautiful pairs of randomizations,
showing in particular that this theory is never -categorical (except
in basic cases).Comment: 28 page
Logic and -algebras: set theoretical dichotomies in the theory of continuous quotients
Given a nonunital -algebra one constructs its corona
algebra . This is the noncommutative analog of the
\v{C}ech-Stone remainder of a topological space. We analyze the two faces of
these algebras: the first one is given assuming CH, and the other one arises
when Forcing Axioms are assumed. In their first face, corona
-algebras have a large group of automorphisms that includes
nondefinable ones. The second face is the Forcing Axiom one; here the
automorphism group of a corona -algebra is as rigid as possible,
including only definable elementsComment: This is the author's Ph.D. thesis, defended in April 2017 at York
University, Toront
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
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