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Godel's theorems and Zermelo's axiom...

Halbeisen, Lorenz.

Godel's theorems and Zermelo's axiomsa firm foundation of mathematics /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Godel's theorems and Zermelo's axiomsby Lorenz Halbeisen, Regula Krapf.
其他題名:	a firm foundation of mathematics /
作者:	Halbeisen, Lorenz.
其他作者:	Krapf, Regula.
出版者:	Cham :Springer International Publishing :2020.
面頁冊數:	xii, 236 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
標題:	Godel's theorem.
電子資源:	https://doi.org/10.1007/978-3-030-52279-7
ISBN:	9783030522797$q(electronic bk.)

Godel's theorems and Zermelo's axiomsa firm foundation of mathematics /
Halbeisen, Lorenz.

Godel's theorems and Zermelo's axiomsa firm foundation of mathematics /[electronic resource] :by Lorenz Halbeisen, Regula Krapf. - Cham :Springer International Publishing :2020. - xii, 236 p. :ill., digital ;24 cm.

A Natural Approach to Natural Numbers -- Part I Introduction to First-Order Logic -- Syntax: The Grammar of Symbols -- Semantics: Making Sense of the Symbols -- Soundness & Completeness -- Part II Godel's Completeness Theorem -- Maximally Consistent Extensions -- Models of Countable Theories -- The Completeness Theorem -- Language Extensions by Definitions -- Part III Godel's Incompleteness Theorems -- Models of Peano Arithmetic and Consequences for Logic -- Arithmetic in Peano Arithmetic -- Godelisation of Peano Arithmetic -- The Incompleteness Theorems -- The Incompleteness Theorems Revisited -- Completeness of Presburger Arithmetic -- Models of Arithmetic Revisited -- Part IV Zermelo's Axioms -- Axioms of Set Theory -- Models of Set Theory -- Models of the Natural and the Real Numbers -- Tautologies.

This book provides a concise and self-contained introduction to the foundations of mathematics. The first part covers the fundamental notions of mathematical logic, including logical axioms, formal proofs and the basics of model theory. Building on this, in the second and third part of the book the authors present detailed proofs of Godel's classical completeness and incompleteness theorems. In particular, the book includes a full proof of Godel's second incompleteness theorem which states that it is impossible to prove the consistency of arithmetic within its axioms. The final part is dedicated to an introduction into modern axiomatic set theory based on the Zermelo's axioms, containing a presentation of Godel's constructible universe of sets. A recurring theme in the whole book consists of standard and non-standard models of several theories, such as Peano arithmetic, Presburger arithmetic and the real numbers. The book addresses undergraduate mathematics students and is suitable for a one or two semester introductory course into logic and set theory. Each chapter concludes with a list of exercises.

ISBN: 9783030522797$q(electronic bk.)

Standard No.: 10.1007/978-3-030-52279-7doiSubjects--Topical Terms:

665760
Godel's theorem.

LC Class. No.: QA9.65 / .H35 2020

Dewey Class. No.: 511.3

Godel's theorems and Zermelo's axiomsa firm foundation of mathematics /
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505 0 $a A Natural Approach to Natural Numbers -- Part I Introduction to First-Order Logic -- Syntax: The Grammar of Symbols -- Semantics: Making Sense of the Symbols -- Soundness & Completeness -- Part II Godel's Completeness Theorem -- Maximally Consistent Extensions -- Models of Countable Theories -- The Completeness Theorem -- Language Extensions by Definitions -- Part III Godel's Incompleteness Theorems -- Models of Peano Arithmetic and Consequences for Logic -- Arithmetic in Peano Arithmetic -- Godelisation of Peano Arithmetic -- The Incompleteness Theorems -- The Incompleteness Theorems Revisited -- Completeness of Presburger Arithmetic -- Models of Arithmetic Revisited -- Part IV Zermelo's Axioms -- Axioms of Set Theory -- Models of Set Theory -- Models of the Natural and the Real Numbers -- Tautologies.
520 $a This book provides a concise and self-contained introduction to the foundations of mathematics. The first part covers the fundamental notions of mathematical logic, including logical axioms, formal proofs and the basics of model theory. Building on this, in the second and third part of the book the authors present detailed proofs of Godel's classical completeness and incompleteness theorems. In particular, the book includes a full proof of Godel's second incompleteness theorem which states that it is impossible to prove the consistency of arithmetic within its axioms. The final part is dedicated to an introduction into modern axiomatic set theory based on the Zermelo's axioms, containing a presentation of Godel's constructible universe of sets. A recurring theme in the whole book consists of standard and non-standard models of several theories, such as Peano arithmetic, Presburger arithmetic and the real numbers. The book addresses undergraduate mathematics students and is suitable for a one or two semester introductory course into logic and set theory. Each chapter concludes with a list of exercises.
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