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Gödel's Theorems and Zermelo's Axioms = A Firm Foundation of Mathematics /

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Gödel's Theorems and Zermelo's Axioms/ by Lorenz Halbeisen, Regula Krapf.
Reminder of title:	A Firm Foundation of Mathematics /
Author:	Halbeisen, Lorenz.
other author:	Krapf, Regula.
Description:	XII, 236 p.online resource. :
Contained By:	Springer Nature eBook
Subject:	Mathematical logic. -
Online resource:	https://doi.org/10.1007/978-3-030-52279-7
ISBN:	9783030522797

Gödel's Theorems and Zermelo's Axioms = A Firm Foundation of Mathematics /
Halbeisen, Lorenz.

Gödel's Theorems and Zermelo's AxiomsA Firm Foundation of Mathematics /[electronic resource] :by Lorenz Halbeisen, Regula Krapf. - 1st ed. 2020. - XII, 236 p.online resource.

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 Gödel’s Completeness Theorem -- Maximally Consistent Extensions -- Models of Countable Theories -- The Completeness Theorem -- Language Extensions by Definitions -- Part III Gödel’s Incompleteness Theorems -- Models of Peano Arithmetic and Consequences for Logic -- Arithmetic in Peano Arithmetic -- Gödelisation 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 Gödel’s classical completeness and incompleteness theorems. In particular, the book includes a full proof of Gödel’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 Gödel’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

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

810627
Mathematical logic.

LC Class. No.: QA8.9-10.3

Dewey Class. No.: 511.3

Gödel's Theorems and Zermelo's Axioms = A 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 Gödel’s Completeness Theorem -- Maximally Consistent Extensions -- Models of Countable Theories -- The Completeness Theorem -- Language Extensions by Definitions -- Part III Gödel’s Incompleteness Theorems -- Models of Peano Arithmetic and Consequences for Logic -- Arithmetic in Peano Arithmetic -- Gödelisation 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 Gödel’s classical completeness and incompleteness theorems. In particular, the book includes a full proof of Gödel’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 Gödel’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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