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
Godel'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."
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