Mathematical logic Books

Book Synopsis- Part I: Personal Notes.- My writing.- Some personal remarks about Johann A. Makowsky.- The Swiss Connection.- From a Friend and Publisher.- From graph polynomials to the software industry Lessons from Janos.- Emancipatory Aspects of Learning and Teaching Mathematics.- Part II: Scientific Contributions.- Epsilon Calculus Provides Shorter Cut-Free Proofs.- Variations on a Theme of Makowsky.- Automatic structures and the problem of natural well-orderings.- On the Counting Complexity of the Cover Polynomial for Simple Graphs.- Polynomial Threshold Functions of Bounded Tree-Width: Some Explainability and Complexity Aspects.- Some Equalities are More Equal than Others.- On the bipartition polynomials for rooted caterpillars.- NP-completeness by first-order and quantifier-free interpretations and related topics.- Bounded languages over infinite alphabets.- Linear Algebraic Quantifiers.- A coarse Tutte polynomial for hypermaps.- Graph polynomials: some questions on the edge.- Pixelating relations and functions without adding substructures.- Reflection and Recurrence.- Provenance Analysis and Semiring Semantics for First-Order Logic.- Reversify any sequential algorithm.- Gentzen in the 3- and 4-valued jungle.-  Characterizing Data Dependencies Then and Now.- On Consistency of Graphically Defined Specifications.- The path-bifurcation hierarchy does not collapse to ??1 in infinite abelian groups.- Data with Logical and Statistical constraints.- Relating Information and Knowledge.- Science and Practice of Modelling.- Graph Polynomials and Local Graph Operations.

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Book SynopsisDieses Buch bietet eine Einführung in die verschiedenen Aspekte der mathematischen Logik. Nach dem Praedikatenkalkül und seinen Anwendungen auf die Anfänge der künstlichen Intelligenz stellt der Autor die Mengenlehre axiomatisch dar. Im dritten und vierten Teil führt er die Grundbegriffe der Berechenbarkeitstheorie und die Hierarchie der Teilmengen ein, um schließlich die Gödelschen Unvollständigkeitssätze zu beweisen. Dieser Band zeichnet sich durch einen klaren Schreibstil aus und enthält zahlreiche Übungsaufgaben.Table of ContentsI Prädikatenkalkül.- 1. Strukturen und Formeln.- 2. Semantik.- 3. Allgemeingültige Formeln.- 4. Der Gödelsche Vollständigkeitssatz.- 5. Der Sequenzenkalkül.- 6. Der Herbrandsche Satz und automatisches Beweisen.- II Mengenlehre.- 7. Die Axiome.- 8. Die natürlichen Zahlen.- 9. Ordinalzahlen und Kardinalzahlen.- 10. Metamathematik von ZFC.- III Rekursionstheorie.- 11. Registermaschinen.- 12. Primitiv rekursive Funktionen und Gödelisierung.- 13. Rekursiv aufzählbare Mengen.- 14. Gödelnummern von Formeln.- 15. Ein anderer Aufbau der rekursiven Funktionen.- IV Arithmetik.- 16. Definierbare Relationen.- 17. Das System Q.- 18. Peanoarithmetik.- 19. Der Zweite Gödelsche Unvollständigkeitssatz.- Literaturverzeichnis.- Index.

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Book SynopsisLectures on intuitionism.- Realizability: A retrospective survey.- Some applications of Kleene's methods for intuitionistic systems.- Notes on intuitionistic second order arithmetic.- Some properties of intuitionistic zermelo-frankel set theory.- Ouelques Resultats sur les Interpretations Fonctionnelles.- Combinator realizability of constructive finite type analysis.- The arithmetic theory of constructions.- The priority method for the construction of recursively enumerable sets.- Admissible ordinals and priority arguments.- Abstract computability versus analog-generability (a survey).- Infinitary combinatorics.- The maximum sum of a family of ordinals.- Effective implications between the "finite" choice axioms.- On descendingly complete ultrafilters.- XVI. A model for the negation of the axiom of choice.- Filters closed under MAHLO's and GAIFMAN's operation.- On chromatic number of graphs and set systems.- Countable models of set theories.- Errata.- Descriptive set theory in .- Modal model theory.- A preservation theorem for interpretations.- Vaught sentences and Lindström's regular relations.Table of ContentsLectures on intuitionism.- Realizability: A retrospective survey.- Some applications of Kleene's methods for intuitionistic systems.- Notes on intuitionistic second order arithmetic.- Some properties of intuitionistic zermelo-frankel set theory.- Ouelques Resultats sur les Interpretations Fonctionnelles.- Combinator realizability of constructive finite type analysis.- The arithmetic theory of constructions.- The priority method for the construction of recursively enumerable sets.- Admissible ordinals and priority arguments.- Abstract computability versus analog-generability (a survey).- Infinitary combinatorics.- The maximum sum of a family of ordinals.- Effective implications between the "finite" choice axioms.- On descendingly complete ultrafilters.- XVI. A model for the negation of the axiom of choice.- Filters closed under MAHLO's and GAIFMAN's operation.- On chromatic number of graphs and set systems.- Countable models of set theories.- Errata.- Descriptive set theory in .- Modal model theory.- A preservation theorem for interpretations.- Vaught sentences and Lindström's regular relations.

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Book SynopsisThe territory of preserver problems has grown continuously within linear analysis. This book presents a cross-section of the modern theory of preservers on infinite dimensional spaces (operator spaces and function spaces) through the author's corresponding results. Special emphasis is placed on preserver problems concerning some structures of Hilbert space operators which appear in quantum mechanics. In addition, local automorphisms and local isometries of operator algebras and function algebras are discussed in detail.Trade ReviewFrom the reviews: "Preserver problems deal with maps on subsets of algebras that preserve certain sets, relations, functions, etc. … The book is very well organized. … One of its remarkable features is that it links several areas, particularly operator theory and mathematical physics. The audience of this book is therefore potentially wide: operator algebraists, mathematical physicists, linear algebraists, ring theorists, etc. I warmly recommend this book to anyone interested in preserver problems." (Matej Brešar, Mathematical Reviews, Issue, 2007 g) "The monograph under review collects many important and highly nontrivial results and efforts. It is important to recall that the basic material is based on the research done by the author, who belongs to the eminent researchers in this field. The style is very fresh … . I recommend to book for students and experts interested in operator algebra, noncommutative measure theory and mathematical foundations of quantum physics. The monograph is welcome in the quantum structures realm." (Anatolij Dvurecenskij, Zentralblatt MATH, Vol. 1119 (21), 2007)Table of ContentsSome Linear and Multiplicative Preserver Problems on Operator Algebras and Function Algebras.- Preservers on Quantum Structures.- Local Automorphisms and Local Isometries of Operator Algebras and Function Algebras.

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Book SynopsisThis book offers a defense against non-classical approaches to the paradoxes. The author argues that, despite appearances, the paradoxes give no reason at all to reject classical logic. In fact, he believes classical solutions fare better than non-classical ones with respect to key tests like Curry’s Paradox, a Liar-like paradox that dialetheists are forced to solve in a way totally disjoint from their solution to the Liar. Graham Priest’s In Contradiction was the first major work that advocated the use of non-classical approaches. Since then, these views have moved into the philosophical mainstream. Much of this movement is fueled by a widespread sense that these logically heterodox solutions get to the real nub of the issue. They lack the ad hoc feel of many other solutions to the paradoxes. The author believes that it's long past time for a response to these attacks against classical orthodoxy. He presents a non-logically-revisionary solution to the paradoxes. This title offers a literal way of cashing out the disquotation metaphor. While the details of the view are novel, the idea has a pre-history in the relevant literature. The author examines objections in detail. He rejects each in turn and concludes by comparing the virtues of his logically orthodox approach with those of the paraconsistent and paracomplete competition.Table of ContentsChapter 1. Logic and the Liar Paradox.- Chapter 2. Graham Priest and Dialetheism.- Chapter 3. How to Solve Priest’s Paradoxes Without Sacrificing Classical Logic.- Chapter 4. Dialetheism and the Laws of Logic.- Chapter 5. Dialetheism, Rejection, and Curry’s Paradox.- Chapter 6. Dialetheism, Rejection, and Probability Theory.- Chapter 7. Hartry Field and Paracompleteness.- Chapter 8. How to Solve the Liar Paradox Without Sacrificing Classical Logic.- Index.

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Book SynopsisDuring his lifetime, Kurt Gödel was not well known outside the professional world of mathematicians, philosophers and theoretical physicists. Early in his career, for his doctoral thesis and then for his Habilitation (Dr.Sci.), he wrote earthshaking articles on the completeness and provability of mathematical-logical systems, upsetting the hypotheses of the most famous mathematicians/philosophers of the time. He later delved into theoretical physics, finding a unique solution to Einstein’s equations for gravity, the ‘Gödel Universe’, and made contributions to philosophy, the guiding theme of his life. This book includes more details about the context of Gödel’s life than are found in earlier biographies, while avoiding an elaborate treatment of his mathematical/scientific/philosophical works, which have been described in great detail in other books. In this way, it makes him and his times more accessible to general readers, and will allow them to appreciate the lasting effects of Gödel’s contributions (the latter in a more up-to-date context than in previous biographies, many of which were written 15–25 years ago). His work spans or is relevant to a wide spectrum of intellectual endeavor, and this is emphasized in the book, with recent examples. This biography also examines possible sources of his unusual personality, which combined mathematical genius with an almost childlike naiveté concerning everyday life, and striking scientific innovations with timidity and hesitancy in practical matters. How he nevertheless had a long and successful career, inspiring many younger scholars along the way, with the help of his loyal wife Adele and some of his friends, is a fascinating story in human nature.Trade Review“The book is generously illustrated, often with color photographs and diagrams, mostly of individuals mentioned in the book, or of maps and buildings where Gödel lived or stayed … at various periods throughout his life. … anyone interested in a general overview of his biography and broad sketches of his technical works and philosophical/theological views will find this an easy read.” (Joseph W. Dauben, Mathematical Reviews, September, 2023)“This biography of Kurt Gödel is written for a wide audience with a limited mathematical background. … The biography is written in a clear and readable style and is very well researched. I was particularly impressed by the author’s familiarity with all things German. … I strongly recommend this biography to anyone interested in the life (and the work) of Kurt Gödel, whose incompleteness theorems spelled the end of Hilbert’s program.” (Franz Lemmermeyer, zbMATH 1505.01001, 2023)“The book is generously illustrated, often with color photographs and diagrams, mostly of individuals mentioned in the book, or of maps and buildings where Gödel lived or stayed … at various periods throughout his life. … anyone interested in a general overview of his biography and broad sketches of his technical works and philosophical/theological views will find this an easy read.” (Joseph W. Dauben, Mathematical Reviews, September, 2023)“This biography of Kurt Gödel is written for a wide audience with a limited mathematical background. … The biography is written in a clear and readable style and is very well researched. I was particularly impressed by the author’s familiarity with all things German. … I strongly recommend this biography to anyone interested in the life (and the work) of Kurt Gödel, whose incompleteness theorems spelled the end of Hilbert’s program.” (Franz Lemmermeyer, zbMATH 1505.01001, 2023)Table of ContentsPreface.- Prologue.- La Belle Époque in Brünn. Beginnings.- School Days. A New Nation.- Moving to the Capital. Student Life in 1920’s Vienna.- The Wiener Kreis and the Mathematical Colloquium. Graduate work.- Private Life in Vienna.- Gödel’s Doctoral Thesis. The Incompleteness Theorems.- The Mathematician in Vienna. Habilitation.- Matters of Health.- A Sojourn Abroad: 1933/34 – Princeton.- Back to Vienna. The First Breakdown.- ‘Commuting’ between Vienna and Princeton – The Late 1930’s. Marriage.- The Continuum Hypothesis.- Professional Uncertainty. A Long Journey Eastwards.- Princeton and the IAS – Philosophy, Einstein and von Neumann. A Bizarre Birthday Present: Gödel‘s Universe.- Reception, Recognition, Honors. Einstein’s Loss. The Professor at Princeton.- The 1960’s: Fame and Seclusion.- Later Years. Philosophy, Cosmology, Logic, Computability.- Gödel’s Legacy – The Lessons of an Unusual Life.- Epilogue.

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Book SynopsisA thorough, self-contained and easily accessible treatment of the theory on the polynomial best approximation of functions with respect to maximum norms. The topics include Chebychev theory, Weierstraß theorems, smoothness of functions, and continuation of functions.Trade Review"The book could be of interest for all who work in approximation theory and related fields; it should not be overlooked by university libraries."In: Ems Newsletter 3/2009 "It is useful for students interested in uniform approximation theory, and it can be used as a reference book for researchers as well."In: L'Enseignement Mathematique 2/2008

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Book SynopsisThis monograph presents recursion theory from a generalized point of view centered on the computational aspects of definability. A major theme is the study of the structures of degrees arising from two key notions of reducibility, the Turing degrees and the hyperdegrees, using techniques and ideas from recursion theory, hyperarithmetic theory, and descriptive set theory. The emphasis is on the interplay between recursion theory and set theory, anchored on the notion of definability. The monograph covers a number of fundamental results in hyperarithmetic theory as well as some recent results on the structure theory of Turing and hyperdegrees. It also features a chapter on the applications of these investigations to higher randomness.Table of ContentsPreliminaries 1. Π11-uniformization and Applications to Turing Degrees 2. Rigidity of Hyperdegrees 3. Basis Theorems and Π11-Hyperarithmetic 4. The Jump Operator 5. Independence Results in the Turing Degrees 6. Higher Randomness References Index

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Book SynopsisOver the last few decades the interest of logicians and mathematicians in constructive and computational aspects of their subjects has been steadily growing, and researchers from disparate areas realized that they can benefit enormously from the mutual exchange of techniques concerned with those aspects. A key figure in this exciting development is the logician and mathematician Helmut Schwichtenberg to whom this volume is dedicated on the occasion of his 70th birthday and his turning emeritus. The volume contains 20 articles from leading experts about recent developments in Constructive set theory, Provably recursive functions, Program extraction, Theories of truth, Constructive mathematics, Classical vs. intuitionistic logic, Inductive definitions, and Continuous functionals and domains.

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Book SynopsisThis comprehensive two-volume work is devoted to the most general beginnings of mathematics. It goes back to Hausdorff’s classic Set Theory (2nd ed., 1927), where set theory and the theory of functions were expounded as the fundamental parts of mathematics in such a way that there was no need for references to other sources. Along the lines of Hausdorff’s initial work (1st ed., 1914), measure and integration theory is also included here as the third fundamental part of contemporary mathematics. The material about sets and numbers is placed in Volume 1 and the material about functions and measures is placed in Volume 2. Contents Historical foreword on the centenary after Felix Hausdorff’s classic Set Theory Fundamentals of the theory of functions Fundamentals of the measure theory Historical notes on the Riesz – Radon – Frechet problem of characterization of Radon integrals as linear functionals

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Book SynopsisThis comprehensive two-volume work is devoted to the most general beginnings of mathematics. It goes back to Hausdorff’s classic Set Theory (2nd ed., 1927), where set theory and the theory of functions were expounded as the fundamental parts of mathematics in such a way that there was no need for references to other sources. Along the lines of Hausdorff’s initial work (1st ed., 1914), measure and integration theory is also included here as the third fundamental part of contemporary mathematics.The material about sets and numbers is placed in Volume 1 and the material about functions and measures is placed in Volume 2. 　 Contents Fundamentals of the theory of classes, sets, and numbers Characterization of all natural models of Neumann – Bernays – Godel and Zermelo – Fraenkel set theories Local theory of sets as a foundation for category theory and its connection with the Zermelo – Fraenkel set theory Compactness theorem for generalized second-order language

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Book SynopsisThe book is about strong axioms of infi nity in set theory (also known as large cardinal axioms), and the ongoing search for natural models of these axioms. Assuming the Ultrapower Axiom, a combinatorial principle conjectured to hold in all such natural models, we solve various classical problems in set theory (for example, the Generalized Continuum Hypothesis) and uncover a theory of large cardinals that is much clearer than the one that can be developed using only the standard axioms.

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Book SynopsisContinuous model theory is an extension of classical first order logic which is best suited for classes of structures which are endowed with a metric. Applications have grown considerably in the past decade. This book is dedicated to showing how the techniques of continuous model theory are used to study C*-algebras and von Neumann algebras. This book geared to researchers in both logic and functional analysis provides the first self-contained collection of articles surveying the many applications of continuous logic to operator algebras that have been obtained in the last 15 years.

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Book SynopsisIn recent years, the interaction between harmonic analysis and convex geometry has increased which has resulted in solutions to several long-standing problems. This collection is based on the topics discussed during the Research Semester on Harmonic Analysis and Convexity at the Institute for Computational and Experimental Research in Mathematics in Providence RI in Fall 2022. The volume brings together experts working in related fields to report on the status of major problems in the area including the isomorphic Busemann-Petty and slicing problems for arbitrary measures, extremal problems for Fourier extension and extremal problems for classical singular integrals of martingale type, among others.

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