Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of (...) the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. (shrink)

We offer a case-study in irrationality, showing that even in a high stakes context, intelligent and well trained professionals may adopt dominated practices. In multiple-choice tests one cannot distinguish lucky guesses from answers based on knowledge. Test-makers have dealt with this problem by lowering the incentive to guess, through penalizing errors (called formula scoring), and by eliminating various cues for outperforming random guessing (e.g., a preponderance of correct answers in middle positions), through key balancing. These policies, though widespread and intuitively (...) appealing, are in fact ‘‘irrational’’, and are dominated by alternative solutions. Number-right scoring is superior to formula scoring, and key randomization is superior to key balancing. We suggest that these policies have persisted since all stake-holders – test-makers, test-takers and test-coaches – share the same faulty intuitions. (shrink)

Yehoshua Bar-Hillel (1915-1975) was one of the leading intellectuals of Israel and of the world. His work ranged over mathematics, applied logic, communication theory, analytic philosophy, philosophy of science, and linguistics. Creative, patient, attentive, and critical, Bar-Hillel was a superb philosopher. In addition, how humane he was may be learned from the memorial tributes to him which initiate this volume. Bar-Hillel was born in Vienna, and came to Israel, then Palestine, in 1933. He took his M. A. (...) (1938) and Ph. D. (1949) at the Hebrew University of Jerusalem, where his subsequent career continued, as Research Fellow (1949-53), Senior Lecturer in Philosophy (1953-58), Associate Professor of Philosophy (1958-61), and Professor of Logic and Philosophy of Sci ence (1961-75). He was often abroad as visiting professor (Berkeley, 1960- 61; Michigan, 1965; La Jolla, 1966-67; Konstanz, 1971; Berlin, 1972), or as a research scholar, notably at the M. lT. Research Laboratory for Elec tronics during the early 1950's. Bar-Hillel was the Secretary and guiding spirit of the Organizing Committee for the 3rd International Congress for Logic, Methodology and Philosophy of Science, held in Jerusalem in 1964. During 1966-68, he was President of the Division of Logic, Method ology and Philosophy of SCience of the International Union of History and Philosophy of Science, and in 1967 President of the International Union. From 1963 he was a Member of the Israel Academy of Sciences and Humanities. (shrink)

One of the tasks with which communication engineers are presented is that of devising a mechanism by which a significant sequence of words, a message, produced by somebody, the sender of the message, is reproduced at some other place, with the shortest practical time lag. The reproduction must be such that the receiver of the message will be able to understand what the sender meant by his message, at least, if he knows the sender's language. The following illustration is typical: (...) A writes on a sheet of paper “I love you” and wishes that B, 3000 miles away, should become aware of the full content of this message, with little delay and at a low cost. There will be institutions, in a capitalistic society, which will compete with each other in providing A, for a price, with the required service. Those companies which perform these services most satisfactorily, i.e., with an overall better combination of faithfulness, time lag, and cost, will get the job. The executives of these companies will hire engineers and put them to work on improving this overall combination. (shrink)