360 research outputs found
On theories of random variables
We study theories of spaces of random variables: first, we consider random
variables with values in the interval , then with values in an arbitrary
metric structure, generalising Keisler's randomisation of classical structures.
We prove preservation and non-preservation results for model theoretic
properties under this construction: i) The randomisation of a stable structure
is stable. ii) The randomisation of a simple unstable structure is not simple.
We also prove that in the randomised structure, every type is a Lascar type
On uniform canonical bases in lattices and other metric structures
We discuss the notion of \emph{uniform canonical bases}, both in an abstract
manner and specifically for the theory of atomless lattices. We also
discuss the connection between the definability of the set of uniform canonical
bases and the existence of the theory of beautiful pairs (i.e., with the finite
cover property), and prove in particular that the set of uniform canonical
bases is definable in algebraically closed metric valued fields
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
Modular functionals and perturbations of Nakano spaces
We settle several questions regarding the model theory of Nakano spaces left
open by the PhD thesis of Pedro Poitevin \cite{Poitevin:PhD}. We start by
studying isometric Banach lattice embeddings of Nakano spaces, showing that in
dimension two and above such embeddings have a particularly simple and rigid
form. We use this to show show that in the Banach lattice language the modular
functional is definable and that complete theories of atomless Nakano spaces
are model complete. We also show that up to arbitrarily small perturbations of
the exponent Nakano spaces are -categorical and -stable. In
particular they are stable
The Algebra of Logic Tradition
The algebra of logic, as an explicit algebraic system showing the underlying mathematical structure of logic, was introduced by George Boole (1815-1864) in his book The Mathematical Analysis of Logic (1847). The methodology initiated by Boole was successfully continued in the 19th century in the work of William Stanley Jevons (1835-1882), Charles Sanders Peirce (1839-1914), Ernst Schröder (1841-1902), among many others, thereby establishing a tradition in (mathematical) logic. From Boole's first book until the influence after WWI of the monumental work Principia Mathematica (1910 1913) by Alfred North Whitehead (1861-1947) and Bertrand Russell (1872-1970), versions of thealgebra of logic were the most developed form of mathematical above allthrough Schröder's three volumes Vorlesungen über die Algebra der Logik(1890-1905). Furthermore, this tradition motivated the investigations of Leopold Löwenheim (1878-1957) that eventually gave rise to model theory. Inaddition, in 1941, Alfred Tarski (1901-1983) in his paper On the calculus of relations returned to Peirce's relation algebra as presented in Schröder's Algebra der Logik. The tradition of the algebra of logic played a key role in thenotion of Logic as Calculus as opposed to the notion of Logic as Universal Language . Beyond Tarski's algebra of relations, the influence of the algebraic tradition in logic can be found in other mathematical theories, such as category theory. However this influence lies outside the scope of this entry, which is divided into 10 sections.Fil: Burris, Stanley. University of Waterloo; CanadáFil: Legris, Javier. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Interdisciplinario de EconomÃa Politica de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Económicas. Instituto Interdisciplinario de EconomÃa Politica de Buenos Aires; Argentin
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