Stephen George Simpson. with definition 1.2.3 and the discussion following it. For example, taking 90(n) to be the formula n §E Y, we have an instance of comprehension, VYEIXVn(n€X<—>n¢Y), asserting that for any given set Y there exists a ...

An introduction to the developments of nonstandard models. Beginning with Godel's incompleteness theorem, it covers the prime models, cofinal extensions, and extensions, Gaifman's construction of a definable type, Tennenbaum's theorem and Friedman's theorem on indicators, ending with a chapter on recursive saturation and resplendency.

This comprehensive monograph is a cornerstone in the area of mathematical logic and related fields. Focusing on Gentzen-type proof theory, the book presents a detailed overview of creative works by the author and other 20th-century logicians that includes applications of proof theory to logic as well as other areas of mathematics. 1975 edition.

People have always been interested in numbers, in particular the natural numbers. Of course, we all have an intuitive notion of what these numbers are. In the late 19th century mathematicians, such as Grassmann, Frege and Dedekind, gave definitions for these familiar objects. Since then the development of axiomatic schemes for arithmetic have played a fundamental role in a logical understanding of mathematics. There has been a need for some time for a monograph on the metamathematics of first-order arithmetic. The (...) aim of the book by Hajek and Pudlak is to cover some of the most important results in the study of a first order theory of the natural numbers, called Peano arithmetic and its fragments (subtheories). The field is quite active, but only a small part of the results has been covered in monographs. This book is divided into three parts. In Part A, the authors develop parts of mathematics and logic in various fragments. Part B is devoted to incompleteness. Part C studies systems that have the induction schema restricted to bounded formulas (Bounded Arithmetic). One highlight of this section is the relation of provability to computational complexity. The study of formal systems for arithmetic is a prerequisite for understanding results such as Gödel's theorems. This book is intended for those who want to learn more about such systems and who want to follow current research in the field. The book contains a bibliography of approximately 1000 items. (shrink)

A theory T is trustworthy iff, whenever a theory U is interpretable in T, then it is faithfully interpretable. In this paper we give a characterization of trustworthiness. We provide a simple proof of Friedman’s Theorem that finitely axiomatized, sequential, consistent theories are trustworthy. We provide an example of a theory whose schematic predicate logic is complete Π20.

Wahrheitstheorien sind ein zentraler Bestandteil von Davidsons semantischem Programm zur Erklärung von Interpretation. Deflationäre Wahrheitskonzeptionen hingegen schreiben dem Wahrheitsprädikat eine minimale explanatorische Funktion zu. Die Frage der Vereinbarkeit dieser beiden Positionen bildet den Kern dieser Untersuchung. Eine Antwort wird durch eine kritische Auseinandersetzung mit Unvereinbarkeitsargumenten und durch eine systematische Betrachtung der Funktion eines deflationären Wahrheitsprädikats anhand von axiomatischen Wahrheitstheorien gegeben. Letztlich wird dafür argumentiert, dass nichts gegen die fruchtbare Anwendbarkeit einer deflationären Wahrheitstheorie innerhalb von Davidsons semantischem Programm spricht.

Some axiomatic theories of truth and related subsystems of second-order arithmetic are surveyed and shown to be conservative over their respective base theory. In particular, it is shown by purely finitistically means that the theory PA ÷ "there is a satisfaction class" and the theory FS of [2] are conservative over PA.

) Modern theories of formal truth have traditionally been used to analyse semantic paradoxes, but their field of application goes well beyond this field to include ontological issues, Godel′s incompleteness phenomena, and the relationship between object language, meta language and reduction. All these fields have had new light sched upon them by studies on the theories of truth. In providing a first summary of the various approaches in this field the author documents their respective advantages and areas of application. The (...) emphasis here is on the axiomatic theories of truth developed by Cantini, Feferman, Friedman and the autor himself, which over the past ten years have increasingly gained ground over the semantic theories of Kripke et al. The book describes the way in which many of the aciomatic proposals are built up and expand on semantic theories. This ist the first time that the results in this field have been made accessible to readers in their larger context. The book deals in detail with the technical aspects and so is ideal as a point of entry into the vast array of literature on the metamathematics of this field. A key objective of the book is to extablish a connection between the latest theories of truth and the older constructions elaborated by Tarski. He made a number of important findings about the formal properties of truth back in the 1930s, but some of his constructions are now considered to be obsolete. Tarski′s views appear in a new light in this book since some of his ideas have only been formulated in recent years. (shrink)

The proof-theoretic results on axiomatic theories oftruth obtained by different authors in recent years are surveyed.In particular, the theories of truth are related to subsystems ofsecond-order analysis. On the basis of these results, thesuitability of axiomatic theories of truth for ontologicalreduction is evaluated.