Richard L. Epstein
This book relates the systems of mathematical logic to their original motivations to formalize reasoning in mathematics. It shows how mathematical logic can be used to formalize particular systems of mathematics and sets out the formalization not only of arithmetic, but also of group theory, field theory, and linear orderings. These lead to the formalization of the real numbers and Euclidean plane geometry. The scope and limitations of modern logic are made clear in these formalizations.
The book provides detailed explanations of all proofs and the insights behind the proofs, as well as detailed and nontrivial examples and problems. The book has more than 550 exercises. It can be used in advanced undergraduate or graduate courses and for self-study and reference.
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"...never gives only the technical side of the matter, but...always offers intuitive motivations and explains basic decisions which constitute the whole approach, and which build a bridge to the students' experiences, with natural language as well as with standard 'elementary' mathematics. The book is a self-contained textbook, requiring as background only some facility in mathematics...This makes the book particularly suitable as a textbook for self-study." —Siegfried J. Gottwald