This is a book review of Cantorian set theory and limitations of size by Michael Hallett.

This book is essential for those interested in Russell, or in the philosophy of logic and mathematics. Only two of the fifteen selections are readily available elsewhere: "On Denoting" and "Mr Strawson on Referring." Amazingly, three substantial selections have never been published previously: "On the Substitutional Theory of Classes and Relations," "The Regressive Method of Discovering the Premisses of Mathematics," and "Is Mathematics Purely Linguistic?". Three other papers have appeared previously only in French : one of these is a better (...) introduction to the theory of types than the classic "Mathematical Logic as based on the Theory of Types." Finally, of the five remaining papers, which are not easily available, three are commentaries on Meinong, one is a note on the axiom of infinity, and the most substantial piece is a discussion of three notions of sets—in effect, Zermelo-Frankel, New Foundations, and Russell’s own "no-classes" theory, which required a theory of types applied to propositional functions. What is striking throughout these essays—all but one from the first decade of the century-is Russell’s persistent sympathy with some sort of realism in foundation theory, and his perception that the theory of definite descriptions, the "no-classes" proposals, and the theory of types, were very closely interrelated. Also included are three bibliographies. The chronological bibliography includes several still unpublished mss in the McMaster University Russell Archives.—J.L. (shrink)

Understanding the Infinite is a loosely connected series of essays on the nature of the infinite in mathematics. The chapters contain much detail, most of which is interesting, but the reader is not given many clues concerning what concepts and ideas are relevant for later developments in the book. There are, however, many technical cross-references, so the reader can expect to spend much time flipping backward and forward.

Reviewed Works:John R. Steel, A. S. Kechris, D. A. Martin, Y. N. Moschovakis, Scales on $\Sigma^1_1$ Sets.Yiannis N. Moschovakis, Scales on Coinductive Sets.Donald A. Martin, John R. Steel, The Extent of Scales in $L$.John R. Steel, Scales in $L$.

Kurt Gödel, mathematician and logician, was one of the most influential thinkers of the twentieth century. Gödel fled Nazi Germany, fearing for his Jewish wife and fed up with Nazi interference in the affairs of the mathematics institute at the University of Göttingen. In 1933 he settled at the Institute for Advanced Study in Princeton, where he joined the group of world-famous mathematicians who made up its original faculty. His 1940 book, better known by its short title, The Consistency of (...) the Continuum Hypothesis, is a classic of modern mathematics. The continuum hypothesis, introduced by mathematician George Cantor in 1877, states that there is no set of numbers between the integers and real numbers. It was later included as the first of mathematician David Hilbert's twenty-three unsolved math problems, famously delivered as a manifesto to the field of mathematics at the International Congress of Mathematicians in Paris in 1900. In The Consistency of the Continuum Hypothesis Gödel set forth his proof for this problem. In 1999, Time magazine ranked him higher than fellow scientists Edwin Hubble, Enrico Fermi, John Maynard Keynes, James Watson, Francis Crick, and Jonas Salk. He is most renowned for his proof in 1931 of the 'incompleteness theorem,' in which he demonstrated that there are problems that cannot be solved by any set of rules or procedures. His proof wrought fruitful havoc in mathematics, logic, and beyond. (shrink)

In this third and last paper on inner models we consider some of the inherent limitations of the method of using inner models of the type defined in 1.2 for the proof of consistency results for the particular system of set theory under consideration. Roughly speaking this limitation may be described by saying that practically no further consistency results can be obtained by the construction of models satisfying the conditions of theorem 1.5, i.e., conditions 1.31, 1.32, 1.33, 1.51, viz.:This applies (...) in particular to the ‘complete models’ defined in 1.4. Before going on to a precise statement of these limitations we shall consider now the theorem on which they depend. This is concerned with a particular type of complete model examples of which we call “proper complete models”; they are those complete models which are essentially interior to the universe, those whose classes are sets of the universe constituting a class thereof, i.e., those for which the following proposition is true:The main theorem of this paper is that the statement that there are no models of this kind can be expressed formally in the same way as the axioms A, B, C and furthermore it can be proved that if the axiom system A, B, C is consistent then so is the system consisting of axioms A, B, C, plus this new hypothesis that there exist no proper complete models. When combined with the axiom ‘V = L’ introduced by Gödel in this new hypothesis yields a system in which any normal complete model which exists has for its universal class V, the universal class of the original system. (shrink)

Sur la décomposition des ensembles de points en parties respectivement congruentes.

Vorlesungen über die Algebra der Logik ist ein unveränderter, hochwertiger Nachdruck der Originalausgabe aus dem Jahr 1890. Hansebooks ist Herausgeber von Literatur zu unterschiedlichen Themengebieten wie Forschung und Wissenschaft, Reisen und Expeditionen, Kochen und Ernährung, Medizin und weiteren Genres. Der Schwerpunkt des Verlages liegt auf dem Erhalt historischer Literatur. Viele Werke historischer Schriftsteller und Wissenschaftler sind heute nur noch als Antiquitäten erhältlich. Hansebooks verlegt diese Bücher neu und trägt damit zum Erhalt selten gewordener Literatur und historischem Wissen auch für die (...) Zukunft bei. (shrink)

This book is a translation of some of the more important parts of the Grundgesetze of Frege: the introduction, the first part of the first volume which gives an exposition of the construction, rules, axioms of Frege's formal system, and two appendices, one of which is from the second volume and gives Frege's analysis of the paradox found by Russell in his system. The editor has provided a long introduction "for those not familiar with Frege," although it will benefit those (...) who have something more than acquaintance with his name as well. Frege's original two-dimensional symbolism has been preserved, but the editor has provided enough discussion in his introduction so as to make the going easier. Next to a complete translation of the Grundgesetze, this work is the most useful introduction to the magnum opus of the nineteenth century's greatest logician.—P. J. M. (shrink)

Provability, Computability and Reflection.