Aspects of incompleteness / Per Lindstrom

The main aim of this book is to present results on incompleteness of arithmetical theories. The book consists of 8 chapters. In Chapter 1, the basic notation and terminology are introduced and a number of relevant facts are stated (usually without proofs). One also finds here the fixed point lemma, the essential undecidability of Robinson's arithmetic Q and the nonexistence of truth definitions. Chapter 2 is devoted to Gödel's incompleteness theorems and their improvements. In Chapter 3 the numerations of recursively enumerable sets are studied. In Chapter 4 one finds results on finite nonaxiomatizability of arithmetical theories. Considerations about conservativity are included in Chapter 5. Chapters 6 and 7 are devoted to mutual interpretability of theories (in particular, in Chapter 7 degrees of interpretability are studied). In Chapter 8 one finds generalizations of results from previous chapters to theories in other languages than the language of arithmetic. Credits for results and proofs, remarks or alternative proofs and related results are given in notes at the ends of the chapters. One also finds some exercises there. The book presupposes some knowledge of and training in mathematical logic, including the completeness theorem and recursion theory. (MathSciNet)