82 research outputs found
Countable Short Recursively Saturated Models of Arithmetic
Short recursively saturated models of arithmetic are exactly the elementary initial segments of recursively saturated models of arithmetic. Since any countable recursively saturated model of arithmetic has continuum many elementary initial segments which are already recursively saturated, we turn our attention to the (countably many) initial segments which are not recursively saturated. We first look at properties of countable short recursively saturated models of arithmetic and show that although these models cannot be cofinally resplendent (an expandability property slightly weaker than resplendency), these models have non-definable expansions which are still short recursively saturated
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
Curious satisfaction classes
We present two new constructions of satisfaction/truth classes over models of
PA (Peano Arithmetic) that provide a foil to the fact that the existence of a
disjunctively correct full truth class over a model M of PA implies that
Con(PA) holds in M.Comment: 12 page
Introduction to Sofic and Hyperlinear groups and Connes' embedding conjecture
Sofic and hyperlinear groups are the countable discrete groups that can be
approximated in a suitable sense by finite symmetric groups and groups of
unitary matrices. These notions turned out to be very deep and fruitful, and
stimulated in the last 15 years an impressive amount of research touching
several seemingly distant areas of mathematics including geometric group
theory, operator algebras, dynamical systems, graph theory, and more recently
even quantum information theory. Several longstanding conjectures that are
still open for arbitrary groups were settled in the case of sofic or
hyperlinear groups. These achievements aroused the interest of an increasing
number of researchers into some fundamental questions about the nature of these
approximation properties. Many of such problems are to this day still open such
as, outstandingly: Is there any countable discrete group that is not sofic or
hyperlinear? A similar pattern can be found in the study of II_1 factors. In
this case the famous conjecture due to Connes (commonly known as the Connes
embedding conjecture) that any II_1 factor can be approximated in a suitable
sense by matrix algebras inspired several breakthroughs in the understanding of
II_1 factors, and stands out today as one of the major open problems in the
field. The aim of these notes is to present in a uniform and accessible way
some cornerstone results in the study of sofic and hyperlinear groups and the
Connes embedding conjecture. The presentation is nonetheless self contained and
accessible to any student or researcher with a graduate level mathematical
background. An appendix by V. Pestov provides a pedagogically new introduction
to the concepts of ultrafilters, ultralimits, and ultraproducts for those
mathematicians who are not familiar with them, and aiming to make these
concepts appear very natural.Comment: 157 pages, with an appendix by Vladimir Pesto
- …