By Godel's incompleteness theorem there is no complete axiomatization of mathematics, not even of first order arithmetic. This leads naturally to the project of investigating the family of, inevitably incomplete, arithmetical theories. In this book we present some of the results obtained in pursuing this aim. A brief summary of the contents of the book is given in the introduction.

Godel's main idea was to translate metamathematical statements, concerning formulas, formal theories, provability in such theories etc., into arithmetic the arithmetization of metamathematics. Combining this with the technique of self-reference, he was able to construct a (formal) sentence which is undecidable, neither provable nor disprovable, in a given theory T of arithmetic, thereby demonstrating, in an entirely novel way, that T is incomplete.

The ideas developed by Godel proved fruitful far beyond their original application. Later essentially new ideas were added. Most important are the basic concepts (and results) of recursion theory these are needed for a completely general formulation (and proof) of Godel's theorem notable are also the so called partial truth-definitions, the idea of formalizing the proof of the completeness of first order logic in arithmetic, and the general concept of interpretability.

In this book we shall be concerned almost exclusively with properties that are common to all sufficiently strong, axiomatizable theories. Thus, for example, investigations of particular theories such as, for example, Peano Arithmetic and its fragments, requiring special, model-theoretic or proof-theoretic, methods, fall outside the scope of the book. But even so, the choice of material in some respects reflects the author's preferences; this is particularly true of Chapter 7.

Credits for results, proofs, and exercises, remarks on alternative proofs, related results, etc. are given in the notes following each chapter. Results (exercises) not attributed to anyone are either easy, folklore, or due to the author.

The reader is assumed to be acquainted with (first order) predicate logic including Henkin's completeness proof. We also presuppose knowledge of the elements of recursion theory. Finally, the reader may find our presentation difficult to follow unless (s)he has previously seen a more detailed development of formal arithmetic, an explicitly defined arithmetization of metamathematics, and a full proof of Godel's theorem.