True arithmetic

Abstract: Preface Background The standard model Discretely ordered rings Godel incompleteness The axioms of Peano arithmetic Some number theory in Peano arithmetic Models of Peano arithmetic Collection Prime models Satisfaction Subsystems of Peano arithmetic Saturation Initial segments The standard system Indicators Recursive saturation Suggestions for further reading Bibliography Index.

Abstract: Recently some interesting first-order statements independent of Peano Arithmetic (P) have been found. Here we present perhaps the first which is, in an informal sense, purely number-theoretic in character (as opposed to metamathematical or combinatorial). The methods used to prove it, however, are combinatorial. We also give another independence result (unashamedly combinatorial in character) proved by the same methods.

Abstract: The MAA is pleased to re-issue the early Carus Mathematical Monographs in ebook and print-on-demand formats. Readers with an interest in the history of the undergraduate curriculum or the history of a particular field will be rewarded by study of these very clear and approachable little volumes. This monograph presents the central ideas of the arithmetic theory of quadratic forms in self-contained form, assuming only knowledge of the fundamentals of matric theory and the theory of numbers. Pertinent concepts of p-adic numbers and quadratic ideals are introduced. It would have been possible to avoid these concepts but the theory gains elegance as well as breadth by the introduction of such relationships. Some results and many of the methods are here presented for the first time. The development begins with the classical theory in the field of reals from the point of view of representation theory, for in these terms many of the later objectives and methods may be revealed. The successive chapters gradually narrow the fields and rings until one has the tools at hand to deal with the classical problems in the ring of rational integers. The analytic theory of quadratic forms is not dealt with because of the delicate analysis involved. However, some of the more important results are stated and references given.