Mathematical Logic and Its Applications 2020

The issue "Mathematical Logic and Its Applications 2020" contains articles related to the following three directions: Descriptive Set Theory (3 articles). Solutions for long-standing problems, including those of A. Tarski and H. Friedman, are presented. Exact combinatorial optimization algorithms, in which the complexity relative to the source data is characterized by a low, or even first degree, polynomial (1 article). III. Applications of mathematical logic and the theory of algorithms (2 articles). The first article deals with the Jacobian and M. Kontsevich’s conjectures, and algorithmic undecidability; for these purposes, non-standard analysis is used. The second article provides a quantitative description of the balance and adaptive resource of a human. Submissions are invited for the next issue "Mathematical Logic and Its Applications 2021

Can you take Akemann--Weaver's ♢\diamondsuit away?

A counterexample to Naimark's problem can be constructed without using Jensen's diamond principle. We also construct, using our weakening of diamond, a separably represented, simple $\textrm{C}^*$-algebra with exactly $m$ inequivalent irreducible representations for all $m\geq 2$. Our principal technical contribution is the introduction of a forcing notion that generically adds an automorphism of a given $\textrm{C}^*$-algebra with a prescribed action on its space of pure states.Comment: 29 page

ON THE FOUNDATIONS OF COMPUTABILITY THEORY

The principal motivation for this work is the observation that there are significant deficiencies in the foundations of conventional computability theory. This thesis examines the problems with conventional computability theory, including its failure to address discrepancies between theory and practice in computer science, semantic confusion in terminology, and limitations in the scope of conventional computing models. In light of these difficulties, fundamental notions are re-examined and revised definitions of key concepts such as “computer,” “computable,” and “computing power” are provided. A detailed analysis is conducted to determine desirable semantics and scope of applicability of foundational notions. The credibility of the revised definitions is ascertained by demonstrating by their ability to address identified problems with conventional definitions. Their practical utility is established through application to examples. Other related issues, including hidden complexity in computations, subtleties related to encodings, and the cardinalities of sets involved in computing, are examined. A resource-based meta-model for characterizing computing model properties is introduced. The proposed definitions are presented as a starting point for an alternate foundation for computability theory. However, formulation of the particular concepts under discussion is not the sole purpose of the thesis. The underlying objective of this research is to open discourse on alternate foundations of computability theory and to inspire re-examination of fundamental notions

Interpretations in Trees with Countably Many Branches

Abstract—We study the expressive power of logical interpreta-tions on the class of scattered trees, namely those with countably many infinite branches. Scattered trees can be thought of as the tree analogue of scattered linear orders. Every scattered tree has an ordinal rank that reflects the structure of its infinite branches. We prove, roughly, that trees and orders of large rank cannot be interpreted in scattered trees of small rank. We consider a quite general notion of interpretation: each element of the interpreted structure is represented by a set of tuples of subsets of the interpreting tree. Our trees are countable, not necessarily finitely branching, and may have finitely many unary predicates as labellings. We also show how to replace injective set-interpretations in (not necessarily scattered) trees by ‘finitary’ set-interpretations. Index Terms—Composition method, finite-set interpretations, infinite scattered trees, monadic second order logic. I