Related papers: Set theory and von Neumann algebras
Metaphysical interpretations of set theory are either inconsistent or incoherent. The uses of sets in mathematics actually involve three distinct kinds of collections (surveyable, definite, and heuristic), which are governed by three…
We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate appropriate to such structures. We indicate the utility of this theory as a framework for the development…
Librationist set theory \pounds ${}$ is developed. It descends from semantics for truth, initiated by Kripke, and others. # extends \pounds, of Librationist closures of the paradoxes in Logic and Logical Philosophy 21(4), 323-361, 2012.…
A new analytic framework is first formalized via the usage of the Monadology (Leibniz 1898), to expand the understanding of Zermelo-Fraenkel-choice set theory (ZFC) and Von Neumann-Bernays-Godel set theory (NBG). Implicitly, the framework…
The algebra generated by the down and up operators on a differential partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on…
In 1996, Michaux and Villemaire considered integer relations $R$ which are not definable in Presburger Arithmetic. That is, not definable in first-order logic over integers with the addition function and the order relation…
We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensaetze) from real algebraic geometry can be generalized to noncommutative $*$-algebras. A…
A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…
The polysymplectic $(n+1)$-form is introduced as an analogue of the symplectic form for the De Donder-Weyl polymomentum Hamiltonian formulation of field theory. The corresponding Poisson brackets on differential forms are constructed. The…
The dimension of any module over an algebra of affiliated operators ${\mathcal U}$ of a finite von Neumann algebra ${\mathcal A}$ is defined using a trace on ${\mathcal A}.$ All zero-dimensional ${\mathcal U}$-modules constitute the torsion…
We develop an axiomatic set theory -- the Theory of Hyperfinite Sets THS, which is based on the idea of existence of proper subclasses of big finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to…
A short proof of a conjecture of Kropholler is given. This gives a relative version of Stallings' Theorem on the structure of groups with more than one end. A generalisation of the Almost Stability Theorem is also obtained, that gives…
This is a brief introduction to the study of growth in groups of Lie type, with $SL_2(\mathbb{F}_q)$ and some of its subgroups as the key examples. They are an edited version of the notes I distributed at the Arizona Winter School in 2016.…
A number of recent papers treated the representation theory of partially ordered sets in unitary spaces with the so called orthoscalar relation. Such theory generalizes the classical theory which studies the representations of partially…
These notes on string theory are based on a series of talks I gave during my graduate studies. As the talks, this introductory essay is intended for young students and non-string theory physicists.
These notes deal with a few aspects of Lie algebras and Lie groups, including some matters related to exponentiation.
Neutrosophic set is a part of neutrosophy which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Neutrosophic set is a powerful general formal framework that has been…
We investigate the oscillator algebra of the Pegg-Barnett oscillator with a finite-dimensional number-state space and show that it possesses the su($n$) Lie algebraic structure. In addition, a so-called supersymmetric Pegg-Barnett…
This paper mainly concerns the von Neumann algebras induced by a tuple of multiplication operators on Bergman spaces which arise essentially from holomorphic proper maps over higher dimensional domains. We study the structures and abelian…
In this paper we develop a structure called Link Algebra, in which we present a Set with two binary operations and an axiom system developed from the study of graph theory and set/antiset theory, sowing main theorems and definitions. Once…