A Model-Theoretic Approach to Proof Theory: 51 (Trends in Logic) - Tapa dura

Libro 41 de 46: Trends in Logic

Kotlarski, Henryk

 
9783030289201: A Model-Theoretic Approach to Proof Theory: 51 (Trends in Logic)
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ISBN 10: 3030289206 ISBN 13: 9783030289201
Editorial: Springer-Verlag GmbH, 2019

Sinopsis

This book presents a detailed treatment of ordinal combinatorics of large sets tailored for independence results. It uses model theoretic and combinatorial methods to obtain results in proof theory, such as incompleteness theorems or a description of the provably total functions of a theory.

In the first chapter, the authors first discusses ordinal combinatorics of finite sets in the style of Ketonen and Solovay. This provides a background for an analysis of subsystems of Peano Arithmetic as well as for combinatorial independence results. Next, the volume examines a variety of proofs of Gödel's incompleteness theorems. The presented proofs differ strongly in nature. They show various aspects of incompleteness phenomena. In additon, coverage introduces some classical methods like the arithmetized completeness theorem, satisfaction predicates or partial satisfaction classes. It also applies them in many contexts. 

The fourth chapter defines the method of indicators for obtaining independence results. It shows what amount of transfinite induction we have in fragments of Peano arithmetic. Then, it uses combinatorics of large sets of the first chapter to show independence results. The last chapter considers nonstandard satisfaction classes. It presents some of the classical theorems related to them. In particular, it covers the results by S. Smith on definability in the language with a satisfaction class and on models without a satisfaction class. 

Overall, the book's content lies on the border between combinatorics, proof theory, and model theory of arithmetic. It offers readers a distinctive approach towards independence results by model-theoretic methods.


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Acerca del autor

Henryk Kotlarski (1949 – 2008) published over forty research articles, most of  them devoted to model theory of Peano arithmetic. He studied nonstandard satisfaction classes, automorphisms of models of Peano arithmetic, clasification of elementary cuts, ordinal combinatorics of finite sets in the style of Ketonen and Solovay, and independence results.

Zofa Adamowicz was a colleague of Henryk Kotlarski for about forty years. They did not write a joint paper but they had a lot of discussions and inspired one another. She shared the main interests of Henryk, in partcular the interest in the incompleteness phenomenon and various proofs of the second Gödel incompleteness theorem.

Teresa Bigorajska is a PhD student of Zofia Adamowicz and a major collaborator of Henryk Kotlarski during his last years. They worked together on ordinal combinatorics of finite sets – a notion heavily used in the book. They studied combinatorial properties of partitions and trees with respect to the notion of largness in the style of Ketonen and Solovay. They developed the machinery for proving independence results presented in the book.

Konrad Zdanowski's research interests focus on theories of arithmetic,  intuitionistic logic, and philosophy. Konrad Zdanowski worked with Henryk Kotlarski on one of his last articles and, through many conversations, he learned from Henryk  some of his approach to arithmetic.

De la contraportada

This book presents a detailed treatment of ordinal combinatorics of large sets tailored for independence results. It uses model theoretic and combinatorial methods to obtain results in proof theory, such as incompleteness theorems or a description of the provably total functions of a theory.

In the first chapter, the authors first discusses ordinal combinatorics of finite sets in the style of Ketonen and Solovay. This provides a background for an analysis of subsystems of Peano Arithmetic as well as for combinatorial independence results. Next, the volume examines a variety of proofs of Gödel's incompleteness theorems. The presented proofs differ strongly in nature. They show various aspects of incompleteness phenomena. In additon, coverage introduces some classical methods like the arithmetized completeness theorem, satisfaction predicates or partial satisfaction classes. It also applies them in many contexts. 

The fourth chapter defines the method of indicators for obtaining independence results. It shows what amount of transfinite induction we have in fragments of Peano arithmetic. Then, it uses combinatorics of large sets of the first chapter to show independence results. The last chapter considers nonstandard satisfaction classes. It presents some of the classical theorems related to them. In particular, it covers the results by S. Smith on definability in the language with a satisfaction class and on models without a satisfaction class. 

Overall, the book's content lies on the border between combinatorics, proof theory, and model theory of arithmetic. It offers readers a distinctive approach towards independence results by model-theoretic methods.


"Sobre este título" puede pertenecer a otra edición de este libro.

Editorial
Springer-Verlag GmbH
Año de publicación
2019
Idioma
Inglés
ISBN 10 
3030289206
ISBN 13 
9783030289201
Encuadernación
Tapa dura
Número de edición
1
Número de páginas
128
Editor
Bigorajska Teresa, Zdanowski Konrad, Adamowicz Zofia
Contacto del fabricante
no disponible
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Otras ediciones populares con el mismo título

9783030289232: A Model-Theoretic Approach to Proof Theory: 51 (Trends in Logic)

Edición Destacada

ISBN 10:  3030289230 ISBN 13:  9783030289232
Editorial: Springer-Verlag GmbH, 2020
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