First-order logic. The origin of modern foundational studies. Frege's system and the paradoxes. The teory of types. Zermelo-Fraenkel set theory. Hilbert's program and Godel's incompleteness theorems. The foundational systems of W.V. Quine. Categorical algebra.

This book presents and survey of the foundations of mathematics. The emphasis is on a mathematical comparison of systems rather than on any exhaustive development of analysis within a single system. Nevertheless, for most systems considered, enough details are given for the development of arithmetic, and the method of constructing the other notions of analysis is indicated. The elements of the general theory of cardinal and ordinal numbers are also furnished in the course of this work.

Chapin reviewed this 1972 ZEITSCHRIFT paper that proves the completeness theorem for the logic of variable-binding-term operators created by Corcoran and his student John Herring in the 1971 LOGIQUE ET ANALYSE paper in which the theorem was conjectured. This leveraging proof extends completeness of ordinary first-order logic to the extension with vbtos. Newton da Costa independently proved the same theorem about the same time using a Henkin-type proof. This 1972 paper builds on the 1971 “Notes on a Semantic Analysis of (...) Variable Binding Term Operators” (Co-author John Herring), Logique et Analyse 55, 646–57. MR0307874 (46 #6989). A variable binding term operator (vbto) is a non-logical constant, say v, which combines with a variable y and a formula F containing y free to form a term (vy:F) whose free variables are exact ly those of F, excluding y. Kalish-Montague 1964 proposed using vbtos to formalize definite descriptions “the x: x+x=2”, set abstracts {x: F}, minimization in recursive function theory “the least x: x+x>2”, etc. However, they gave no semantics for vbtos. Hatcher 1968 gave a semantics but one that has flaws described in the 1971 paper and admitted by Hatcher. In 1971 we give a correct semantic analysis of vbtos. We also give axioms for using them in deductions. And we conjecture strong completeness for the deductions with respect to the semantics. The conjecture, proved in this paper with Hatcher’s help, was proved independently about the same time by Newton da Costa. (shrink)

The late Dr. William Hatcher was renowned both as an accomplished mathematician, logician and philosopher, and as one of the world?s foremost Bahá?í scholars. To those who knew him, he was also refreshingly approachable, supportive, honest and engaging. In this small but richly insightful volume, modern mathematics and logic meet religion and philosophy in a new and stimulating way. The five essays are entitled: "Platonism and Pragmatism", "Myths, Models and Mysticism", "From Metaphysics to Logic", "A Logical Solution to the Problem (...) of Evil", and "Science and the Bahá?í Faith" With characteristic originality, Dr. Hatcher explores the interface between these disciplines now approaching each other after centuries of mutual mistrust. These valuable intellectual contributions assist the reader to bring into closer harmony the two ultimate realms expressive of essential human nature - namely that of science and of religion. (shrink)

With fascinating insight, Love, Power, and Justice explores issues of authentic morality using precepts and arguments from philosophy, science and religion, as well as the profound concepts contained in the Baha'i revelation. This work, now in its second edition, is an innovative contribution to one of the more intractable debates of our time--a time when so many different factions and individuals each claim to speak with moral authority.