Mathematical Logic and Its and Mathematical Logic Applications 2020 • Vassily Lyubetsky and Vladimir Kanovei Mathematical Logic and Its Applications 2020 Edited by Vassily Lyubetsky and Vladimir Kanovei Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Mathematical Logic and Its Applications 2020 Mathematical Logic and Its Applications 2020 Editors Vassily Lyubetsky Vladimir Kanovei MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editors Vassily Lyubetsky Vladimir Kanovei Department of Mechanics and Institute for Information Mathematics of Moscow Transmission Problems of the Lomonosov State University Russian Academy of Sciences Russia Russia Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) (available at: https://www.mdpi.com/journal/mathematics/special issues/math-logic-2020). 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Contents About the Editors .............................................. vii Preface to ”Mathematical Logic and Its Applications 2020” ..................... ix Vladimir Kanovei and Vassily Lyubetsky Models of Set Theory in which NonconstructibleReals First Appear at a Given Projective Level Reprinted from: Mathematics 2020, 8, 910, doi:10.3390/math8060910 ................. 1 Vladimir Kanovei and Vassily Lyubetsky 1 On the Δn Problem of Harvey Friedman Reprinted from: Mathematics 2020, 8, 1477, doi:10.3390/math8091477 ................ 47 Vladimir Kanovei and Vassily Lyubetsky On the ‘Definability of Definable’Problem of Alfred Tarski Reprinted from: Mathematics 2020, 8, 2214, doi:10.3390/math8122214 ................ 77 Konstantin Gorbunov and Vassily Lyubetsky Linear Time Additively Exact Algorithm for Transformation of Chain-Cycle Graphs for Arbitrary Costs of Deletions and Insertions Reprinted from: Mathematics 2020, 8, 2001, doi:10.3390/math8112001 ................113 Alexei Kanel-Belov, Alexei Chilikov, Ilya Ivanov-Pogodaev, Sergey Malev, Eugeny Plotkin, Jie-Tai Yu and Wenchao Zhang Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry Reprinted from: Mathematics 2020, 8, 1694, doi:10.3390/math8101694 ................143 Irina Alchinova and Mikhail Karganov Physiological Balance of the Body: Theory, Algorithms, and Results Reprinted from: Mathematics 2021, 9, 209, doi:10.3390/math9030209 .................177 v About the Editors Vassily Lyubetsky is a Doctor of physical and mathematical sciences, professor, and a principal researcher. He is also head of the Laboratory of Mathematical Methods and Models in Bioinformatics at the Institute for Information Transmission Problems of the Russian Academy of Sciences (IITP, Moscow), and Professor of the Faculty of Mechanics and Mathematics of the Moscow State University of the Department of Mathematical Logic and Theory of Algorithms. He has published more than 200 scientific papers and 9 books, and the Guest Invited Editor for the Regular Special Issue ”Molecular Phylogenomics” of the ”Biomed Research International journal (Molecular Biology)”. Vladimir Kanovei is a principal researcher at the Institute for Information Transmission Problems of the Russian Academy of Sciences (IITP, Moscow). He was awarded his Ph.D. by the Moscow State University, and his D.Sc. by the Steklov Mathematical Institute of the Russian Academy of Sciences. His research interests in mathematics include mathematical logic, descriptive set theory, forcing, and nonstandard analysis. He is an author of about 300 scientific publications, including 7 monographs. vii Preface to ”Mathematical Logic and Its Applications 2020” This Special Issue contains articles representing three directions: Descriptive set theory (DTM), exact polynomial complexity algorithms (EPA), and applications of mathematical logic and algorithm theory (Appl). We will say a few words about each of the directions. In accordance with the classical description of Nicolas Luzin, DTM considers simple properties of simple sets of real numbers R. “Simple” sets are Borel sets (the smallest family containing n open and closed sets in R and closed with respect to the operations of countable union and countable intersection) and projective sets (the smallest family containing Borel sets and closed n m with respect to the operations of projecting from R to R , m<n, and the complement to the whole space). The question of what is a “simple” property is more complicated, but it is not important, since in fact we study a small list of individual properties, including the 1 Lebesgue measurability, Baire property , and the individual definability of a set, function, or real. The latest means that there is a formula that holds for a given real number and for no others. This depends on the class of formulas allowed. Such a natural class consists of formulas of the form ∀x1 ∃y1 ∀x2 ∃y2 ... ∀xn ∃yn ψ(x1,y1,...,xn,yn,x), where the variables x1,y1,...,xn,yn,x run through the whole R, and the elementary part ψ(x1,y1,...,xn,yn,x) is any arithmetic formula (which contains any quantifiers over the natural numbers, as well as equalities and inequalities that connect the superpositions of operations from the semiring of natural numbers). To date, the development of DTM leads to a non-trivial general cultural conclusion: every real number is definable (using countable 2 ordinals )orrandom; in the latter case it does not possess any non-trivial properties. This implies that 3 there are absolutely undecidable statements ; as well as surprising connections between seemingly very different absolutely undecidable ones. For example, the measurability implies the Baire property for a wide class of sets. The first three articles belong to this direction. In particular, they solve the well-known problem (1948) of A. Tarski on the definability of the notion of definability itself, and prove the statement (1975) of H. Friedman. The EPA section contains an article contributing a solution for the meaningful combinatorial and, at first glance, complicated algorithmic problem of optimization of the functional given on paths of passing from one graph to another. It is solved by an algorithm of linear complexity, being at the same time exact. The latter means that for any input data, that is for any ordered pair of graphs A and B, accompanied by costs of elementary graph transformations, the algorithm produces exactly the minimal value of the above functional (i.e., the minimum distance between A and B and the minimum path itself from A to B). Here the complexity of the problem turned into the logical complexity of this, albeit linear, algorithm. Our goal was to draw attention to the search for, and possible discussion of, algorithmic problems that seem to require exhaustive search but are actually solved by exact algorithms of low polynomial complexity. This ensures their practical significance when working with large data (terabyte and larger sizes). The Appl section contains two articles. First of them is devoted to the application of non-standard analysis (and other logical methods) to the problems of isomorphism in algebra and mathematical physics (the Jacobian and M. Kontsevich’s conjectures, and algorithmic undecidability). The second is devoted to the application of logical and algorithmic approaches to the problem of theoretical medicine — a quantitative description of the balance and the adaptive resource of a human ix that determines his resistance to external influences. Applied problems in which logic and theory of algorithms have shown their usefulness could be of interest. The Editorial Board of Mathematics (WoS: Q1) has announced the preparation of the issue “Mathematical Logic and Its Applications 2021”; contributions in these directions and especially in other ones of this huge mathematical area, including various applications, are invited. 1) The Baire property of a set X says that there is an open set U such that the symmetric difference X Δ U is a meager set (the union of a countable number of nowhere-dense sets). 2) Countable ordinals are the natural numbers themselves and their natural extension: taking the limit over all natural numbers we get ω, adding +1 to it consecutively and taking the limit yet again we get ω + ω = ω · 2, and so on. Each time, the limit is taken over a countable sequence. 3) This means that a natural statement about measurability (or other simple subjects) cannot be proved or disproved in the natural set theory of ZFC, which seems to contain all the mathematics used in physics, biology, computer science, and engineering. Vassily Lyubetsky, Vladimir Kanovei Editors x mathematics Article Models of Set Theory in which Nonconstructible Reals First Appear at a Given