is the second-order theory of the part-whole relation. It can express such hypotheses about the size of Reality as that there are inaccessibly many atoms. Take a non-empty class to have exactly its non-empty subclasses as parts; hence, its singleton subclasses as atomic parts. Then standard set theory becomes the theory of the member-singleton function—better, the theory of all singleton functions—within the framework of megethology. Given inaccessibly many atoms and a specification of which atoms are urelements, a singleton function exists, (...) unique up to isomorphism. (This article is partly abridged from my Parts of Classes, partly a sequel.). (shrink)

NOWADAYS, even schoolchildren babble about "null sets" and "singletons" and "one-one correspondences," as if they knew what they were talking about. But if they understand even less than their teachers, which seems likely, they must be using the technical jargon with only an illusion of understanding. Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should (...) be child's play. (shrink)

This book, written by one of philosophy's pre-eminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. It is therefore a book of critical importance to logical theory. Jaakko Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. The famous (...) impossibility results by Gödel and Tarski that have dominated the field for the last sixty years turn out to be much less significant than has been thought. All of ordinary mathematics can in principle be done on this first-order level, thus dispensing with the existence of sets and other higher-order entities. (shrink)