This book appeared about ten years ago in Gennan. It started as notes for a course which I gave intermittently at the ETH over a number of years. Following repeated suggestions, this English translation was commissioned by Springer; they were most fortunate in finding translators whose mathematiยญ cal stature, grasp of the language and unselfish dedication to the essentially thankless task of rendering the text comprehensible in a second language, both impresses and shames me. Therefore, my thanks go to Dr. Roberto Minio, now Darmstadt and Professor Charles Thomas, Cambridge. The task of preparing a La'JEX-version of the text was extremely daunting, owing to the complexity and diversity of the symbolisms inherent in the various parts of the book. Here, my warm thanks go to Barbara Aquilino of the Mathematics Department of the ETH, who spent tedious but exacting hours in front of her Olivetti. The present book is not primarily intended to teach logic and axiomatยญ ics as such, nor is it a complete survey of what was once called "elementary mathematics from a higher standpoint". Rather, its goal is to awaken a certain critical attitude in the student and to help give this attitude some solid founยญ dation. Our mathematics students, having been drilled for years in high-school and college, and having studied the immense edifice of analysis, regrettably come away convinced that they understand the concepts of real numbers, Euclidean space, and algorithm
CONTENT
I. The Continuum -- ยง1 What Are the Real Numbers? -- ยง2 Language as Part of Mathematics -- ยง3 Elementary Theory of Real Numbers -- ยง4 Non-Standard Analysis -- ยง5 Axiom of Choice and Continuum Hypothesis -- II. Geometry -- ยง1 Space and Mathematics -- ยง2 Axiomatization by Means of Coordinates -- ยง3 Metatheoretical Questions and Methods in Elementary Geometry -- ยง4 Geometric Constructions -- III. Algorithmics -- ยง1 What is an Algorithm? -- ยง2 The Existence of Combinatory Algebras: Combinatory Logic -- ยง3 Concrete Combinatory Algebras -- ยง4 Lambda Calculus -- ยง5 Computability and Combinatorss
