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Author : Petr Hájek,Pavel Pudlák
Publisher : Cambridge University Press
Page : 475 pages
File Size : 46,9 Mb
Release : 2017-03-02
Category : Mathematics
ISBN : 9781107168411
A much-needed monograph on the metamathematics of first-order arithmetic, paying particular attention to fragments of Peano arithmetic.
Author : Petr Hájek,Pavel Pudlák
Publisher : Cambridge University Press
Page : 476 pages
File Size : 47,7 Mb
Release : 2017-03-02
Category : Mathematics
ISBN : 9781316739457
Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the third publication in the Perspectives in Logic series, is a much-needed monograph on the metamathematics of first-order arithmetic. The authors pay particular attention to subsystems (fragments) of Peano arithmetic and give the reader a deeper understanding of the role of the axiom schema of induction and of the phenomenon of incompleteness. The reader is only assumed to know the basics of mathematical logic, which are reviewed in the preliminaries. Part I develops parts of mathematics and logic in various fragments. Part II is devoted to incompleteness. Finally, Part III studies systems that have the induction schema restricted to bounded formulas (bounded arithmetic).
Author : Petr Hajek,Pavel Pudlak
Publisher : Springer
Page : 460 pages
File Size : 52,9 Mb
Release : 1998-03-17
Category : Mathematics
ISBN : 354063648X
People have always been interested in numbers, in particular the natural numbers. Of course, we all have an intuitive notion of what these numbers are. In the late 19th century mathematicians, such as Grassmann, Frege and Dedekind, gave definitions for these familiar objects. Since then the development of axiomatic schemes for arithmetic have played a fundamental role in a logical understanding of mathematics. There has been a need for some time for a monograph on the metamathematics of first-order arithmetic. The aim of the book by Hajek and Pudlak is to cover some of the most important results in the study of a first order theory of the natural numbers, called Peano arithmetic and its fragments (subtheories). The field is quite active, but only a small part of the results has been covered in monographs. This book is divided into three parts. In Part A, the authors develop parts of mathematics and logic in various fragments. Part B is devoted to incompleteness. Part C studies systems that have the induction schema restricted to bounded formulas (Bounded Arithmetic). One highlight of this section is the relation of provability to computational complexity. The study of formal systems for arithmetic is a prerequisite for understanding results such as Gödel's theorems. This book is intended for those who want to learn more about such systems and who want to follow current research in the field. The book contains a bibliography of approximately 1000 items.
Author : P Hajek
Publisher : Unknown
Page : 128 pages
File Size : 55,5 Mb
Release : 1998
Category : Electronic
ISBN : OCLC:1316614347
Author : Robert L. Rogers
Publisher : Elsevier
Page : 248 pages
File Size : 46,8 Mb
Release : 2014-05-12
Category : Mathematics
ISBN : 9781483257976
Mathematical Logic and Formalized Theories: A Survey of Basic Concepts and Results focuses on basic concepts and results of mathematical logic and the study of formalized theories. The manuscript first elaborates on sentential logic and first-order predicate logic. Discussions focus on first-order predicate logic with identity and operation symbols, first-order predicate logic with identity, completeness theorems, elementary theories, deduction theorem, interpretations, truth, and validity, sentential connectives, and tautologies. The text then tackles second-order predicate logic, as well as second-order theories, theory of definition, and second-order predicate logic F2. The publication takes a look at natural and real numbers, incompleteness, and the axiomatic set theory. Topics include paradoxes, recursive functions and relations, Gödel's first incompleteness theorem, axiom of choice, metamathematics of R and elementary algebra, and metamathematics of N. The book is a valuable reference for mathematicians and researchers interested in mathematical logic and formalized theories.
Author : Stephen Cole Kleene
Publisher : Unknown
Page : 568 pages
File Size : 50,7 Mb
Release : 1971
Category : Functions
ISBN : UCSD:31822013069018
Author : Roman Murawski
Publisher : Springer Science & Business Media
Page : 416 pages
File Size : 52,6 Mb
Release : 2013-03-14
Category : Philosophy
ISBN : 9789401728669
Recursive Functions and Metamathematics deals with problems of the completeness and decidability of theories, using as its main tool the theory of recursive functions. This theory is first introduced and discussed. Then Gödel's incompleteness theorems are presented, together with generalizations, strengthenings, and the decidability theory. The book also considers the historical and philosophical context of these issues and their philosophical and methodological consequences. Recent results and trends have been included, such as undecidable sentences of mathematical content, reverse mathematics. All the main results are presented in detail. The book is self-contained and presupposes only some knowledge of elementary mathematical logic. There is an extensive bibliography. Readership: Scholars and advanced students of logic, mathematics, philosophy of science.
Author : N. Shankar
Publisher : Cambridge University Press
Page : 224 pages
File Size : 52,6 Mb
Release : 1997-01-30
Category : Computers
ISBN : 0521585333
Describes the use of computer programs to check several proofs in the foundations of mathematics.
Author : Yong Cheng
Publisher : Springer Nature
Page : 122 pages
File Size : 44,5 Mb
Release : 2019-08-30
Category : Mathematics
ISBN : 9789811399497
Gödel's true-but-unprovable sentence from the first incompleteness theorem is purely logical in nature, i.e. not mathematically natural or interesting. An interesting problem is to find mathematically natural and interesting statements that are similarly unprovable. A lot of research has since been done in this direction, most notably by Harvey Friedman. A lot of examples of concrete incompleteness with real mathematical content have been found to date. This brief contributes to Harvey Friedman's research program on concrete incompleteness for higher-order arithmetic and gives a specific example of concrete mathematical theorems which is expressible in second-order arithmetic but the minimal system in higher-order arithmetic to prove it is fourth-order arithmetic. This book first examines the following foundational question: are all theorems in classic mathematics expressible in second-order arithmetic provable in second-order arithmetic? The author gives a counterexample for this question and isolates this counterexample from the Martin-Harrington Theorem in set theory. It shows that the statement “Harrington's principle implies zero sharp" is not provable in second-order arithmetic. This book further examines what is the minimal system in higher-order arithmetic to prove the theorem “Harrington's principle implies zero sharp" and shows that it is neither provable in second-order arithmetic or third-order arithmetic, but provable in fourth-order arithmetic. The book also examines the large cardinal strength of Harrington's principle and its strengthening over second-order arithmetic and third-order arithmetic.
Author : Alfred North Whitehead,Bertrand Russell
Publisher : Cambridge University Press
Page : 524 pages
File Size : 49,7 Mb
Release : 1927
Category : Mathematics
ISBN : 052106791X
The Principia Mathematica has long been recognised as one of the intellectual landmarks of the century.
Author : Stephen George Simpson
Publisher : Cambridge University Press
Page : 461 pages
File Size : 50,6 Mb
Release : 2009-05-29
Category : Mathematics
ISBN : 9780521884396
This volume examines appropriate axioms for mathematics to prove particular theorems in core areas.
Author : Raymond M. Smullyan
Publisher : Oxford University Press
Page : 184 pages
File Size : 49,5 Mb
Release : 1993-01-28
Category : Mathematics
ISBN : 0195344812
This work is a sequel to the author's Gödel's Incompleteness Theorems, though it can be read independently by anyone familiar with Gödel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new results in the field.
Author : Mark Burgin
Publisher : World Scientific
Page : 370 pages
File Size : 55,9 Mb
Release : 2022-04-22
Category : Mathematics
ISBN : 9789811236853
The book is the first in the trilogy which will bring you to the fascinating world of numbers and operations with them. Numbers provide information about myriads of things. Together with operations, numbers constitute arithmetic forming in basic intellectual instruments of theoretical and practical activity of people and offering powerful tools for representation, acquisition, transmission, processing, storage, and management of information about the world.The history of numbers and arithmetic is the topic of a variety of books and at the same time, it is extensively presented in many books on the history of mathematics. However, all of them, at best, bring the reader to the end of the 19th century without including the developments in these areas in the 20th century and later. Besides, such books consider and describe only the most popular classes of numbers, such as whole numbers or real numbers. At the same time, a diversity of new classes of numbers and arithmetic were introduced in the 20th century.This book looks into the chronicle of numbers and arithmetic from ancient times all the way to 21st century. It also includes the developments in these areas in the 20th century and later. A unique aspect of this book is its information orientation of the exposition of the history of numbers and arithmetic.
Author : Elliott Mendelson
Publisher : CRC Press
Page : 499 pages
File Size : 46,9 Mb
Release : 2015-05-21
Category : Mathematics
ISBN : 9781482237788
The new edition of this classic textbook, Introduction to Mathematical Logic, Sixth Edition explores the principal topics of mathematical logic. It covers propositional logic, first-order logic, first-order number theory, axiomatic set theory, and the theory of computability. The text also discusses the major results of Godel, Church, Kleene, Rosse
Author : Franco Montagna
Publisher : Springer
Page : 318 pages
File Size : 40,5 Mb
Release : 2014-09-23
Category : Mathematics
ISBN : 9783319062334
This volume celebrates the work of Petr Hájek on mathematical fuzzy logic and presents how his efforts have influenced prominent logicians who are continuing his work. The book opens with a discussion on Hájek's contribution to mathematical fuzzy logic and with a scientific biography of him, progresses to include two articles with a foundation flavour, that demonstrate some important aspects of Hájek's production, namely, a paper on the development of fuzzy sets and another paper on some fuzzy versions of set theory and arithmetic. Articles in the volume also focus on the treatment of vagueness, building connections between Hájek's favorite fuzzy logic and linguistic models of vagueness. Other articles introduce alternative notions of consequence relation, namely, the preservation of truth degrees, which is discussed in a general context, and the differential semantics. For the latter, a surprisingly strong standard completeness theorem is proved. Another contribution also looks at two principles valid in classical logic and characterize the three main t-norm logics in terms of these principles. Other articles, with an algebraic flavour, offer a summary of the applications of lattice ordered-groups to many-valued logic and to quantum logic, as well as an investigation of prelinearity in varieties of pointed lattice ordered algebras that satisfy a weak form of distributivity and have a very weak implication. The last part of the volume contains an article on possibilistic modal logics defined over MTL chains, a topic that Hájek discussed in his celebrated work, Metamathematics of Fuzzy Logic, and another one where the authors, besides offering unexpected premises such as proposing to call Hájek's basic fuzzy logic HL, instead of BL, propose a very weak system, called SL as a candidate for the role of the really basic fuzzy logic. The paper also provides a generalization of the prelinearity axiom, which was investigated by Hájek in the context of fuzzy logic.
