Elliott Mendelson

The more traditional approaches to the history and philosophy of science and technology continue as well, and probably will continue as long as there are skillful practitioners such as Carl Hempel, Ernest Nagel, and th~ir students. Finally, there are still other approaches that address some of the technical problems arising when we try to provide an account of belief and of rational choice. - These include efforts to provide logical frameworks within which we can make sense of these notions. This series will attempt to bring together work from all of these approaches to the history and philosophy of science and technology in the belief that each has something to add to our understanding. The volumes of this series have emerged either from lectures given by authors while they served as honorary visiting professors at the City College of New York or from conferences sponsored by that institution. The City College Program in the History and Philosophy of Science and Technology oversees and directs these lectures and conferences with the financial aid of the Association for Philosophy of Science, Psychotheraphy, and Ethics. MARTIN TAMNY RAPHAEL STERN PREFACE The papers in this collection stem largely from the conference 'Foun dations: Logic, Language, and Mathematics' held at the Graduate Center of the City University of New York on 14-15 November 1980.

The author tells us that this book was originally intended to be part of a larger work, with the provisional title God and Infinity, but that he opted instead for a separate and independent treatment of the notion of infinity in philosophy and related areas. The original purpose is very well-hidden, showing itself clearly only in the Preface and a few other places. The book begins with a chapter describing some known alleged difficulties having to do with the infinitely large and infinitely small . This is followed by a chapter on some basic mathematical notions and facts, including sketchy references to Zermelo-Fraenkel set theory , ordinal and cardinal numbers, Cantor's diagonal method, some special set-theoretic statements , analysis, non-standard analysis, model theory, internal set theory , Conway numbers, and the ‘finite mathematics’ of Mycielski and Lavine. Of course, readers will obtain very little solid information about these matters, but at least they will be introduced to some of the terminology. Chapter 3 returns to the problems and conundrums introduced in Chapter 1 and tries to resolve them or show that they are illusory. In some cases, the trouble arose because an infinite sequence of experiments was imagined which would be physically impossible . Other apparent difficulties arose only because of unfamiliarity with the …