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A model-theoretic approach to proof ...

Adamowicz, Zofia.

A model-theoretic approach to proof theory

Record Type:	Electronic resources : Monograph/item
Title/Author:	A model-theoretic approach to proof theoryby Henryk Kotlarski ; edited by Zofia Adamowicz, Teresa Bigorajska, Konrad Zdanowski.
Author:	Kotlarski, Henryk.
other author:	Adamowicz, Zofia.
Published:	Cham :Springer International Publishing :2019.
Description:	xviii, 109 p. :ill., digital ;24 cm.
Contained By:	Springer eBooks
Subject:	Proof theory.
Online resource:	https://doi.org/10.1007/978-3-030-28921-8
ISBN:	9783030289218$q(electronic bk.)

A model-theoretic approach to proof theory
Kotlarski, Henryk.

A model-theoretic approach to proof theory[electronic resource] /by Henryk Kotlarski ; edited by Zofia Adamowicz, Teresa Bigorajska, Konrad Zdanowski. - Cham :Springer International Publishing :2019. - xviii, 109 p. :ill., digital ;24 cm. - Trends in logic, studia logica library,v.511572-6126 ;. - Trends in logic, studia logica library ;v.41..

Chapter 1. Some combinatorics -- Chapter 2. Some model theory -- Chapter 3. Incompleteness -- Chapter 4. Transﬁnite induction -- Chapter 5. Satisfaction classes.

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 Godel'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.

ISBN: 9783030289218$q(electronic bk.)

Standard No.: 10.1007/978-3-030-28921-8doiSubjects--Topical Terms:

190840
Proof theory.

LC Class. No.: QA9.54 / .K685 2019

Dewey Class. No.: 511.36

A model-theoretic approach to proof theory
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245 1 2 $a A model-theoretic approach to proof theory $h [electronic resource] / $c by Henryk Kotlarski ; edited by Zofia Adamowicz, Teresa Bigorajska, Konrad Zdanowski.
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490 1 $a Trends in logic, studia logica library, $x 1572-6126 ; $v v.51
505 0 $a Chapter 1. Some combinatorics -- Chapter 2. Some model theory -- Chapter 3. Incompleteness -- Chapter 4. Transﬁnite induction -- Chapter 5. Satisfaction classes.
520 $a 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 Godel'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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