This book provides a concise and self-contained introduction to the
foundations of mathematics. The first part covers the fundamental
notions of mathematical logic, including logical axioms, formal
proofs and the basics of model theory. Building on this, in the
second and third part of the book the authors present detailed
proofs of Goedel's classical completeness and incompleteness
theorems. In particular, the book includes a full proof of Goedel's
second incompleteness theorem which states that it is impossible to
prove the consistency of arithmetic within its axioms. The final
part is dedicated to an introduction into modern axiomatic set
theory based on the Zermelo's axioms, containing a presentation of
Goedel's constructible universe of sets. A recurring theme in the
whole book consists of standard and non-standard models of several
theories, such as Peano arithmetic, Presburger arithmetic and the
real numbers. The book addresses undergraduate mathematics students
and is suitable for a one or two semester introductory course into
logic and set theory. Each chapter concludes with a list of
exercises.
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