The Continuum: A Constructive Approach to Basic Concepts of Real Analysis [Paperback]

In this small text the basic theory of the continuum, including the elements of metric space theory and continuity is developed within the system of intuitionistic mathematics in the sense of L.E.J. Brouwer and H. Weyl. The main features are proofs of the famous theorems of Brouwer concerning the continuity of all functions that are defined on whole intervals, the uniform continuity of all functions that are defined on compact intervals, and the uniform convergence of all pointwise converging sequences of functions defined on compact intervals. The constructive approach is interesting both in itself and as a contrast to, for example, the formal axiomatic one. Few mathematical structures have undergone as many revlSlons or have been presented in as many guises as the real numbers. Every generation re-examines the reals in the light of its values and mathematical objectives. This citation is said to be due to Gian-Carlo Rota, and in this book its correctness again is affirmed. Here I propose to investigate the structure of the mathematical continuum by undertaking a rather unconventional access to the real numbers: the intuitionistic one. The traces can be tracked back at least to L.E.J. Brouwer and to H. Weyl. Largely unknown photographies of Weyl in Switzerland after World War II provided by Peter Bettschart enliven the abstract text full of subtle definitions and sophisticated estimations. The book can be read by students who have undertaken the usual analysis courses and want to know more about the intrinsic details of the underlying concepts, and it can also be used by university teachers in lectures for advanced undergraduates and in seminaries for graduate students. I wish to thank Walter Lummerding and Gottfried Oehl who helped me with their impressive expert knowledge of the English language. I also take the opportunity to express my gratitude to Ulrike Schmickler-Hirzebruch and to the staff of Vieweg-Verlag for editing my manuscript just now, exactly 50 ylc"