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.
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.
