Background
Cantor, Georg was born on April 3, 1845 in St Petersburg, Russia.
Cantor, Georg was born on April 3, 1845 in St Petersburg, Russia.
University of Berlin.
Professor, University of Halle, 1867-1913.
Main publications:
(1932) Gesummelte Abhandlungen. Berlin
Springer.
Secondary literature:
Dauben, J. W. (1979) Georg Cantor, Cambridge, Mass.: Harvard University Press.
Fraenkel, A. (1953) Abstract Set Theory, Amsterdam: North-Holland.
It is largely due to the work of Cantor that set theory and transfinite arithmetic have come to play such important roles in mathematics, logic and some aspects of philosophy. His own philosophical foundation of his theories, however, was not well received.
Cantor started to develop these theories in the early 1870s, in connection with solving certain problems in mathematical analysis of the time. Over the next twenty-five years he gradually developed them: both point set topology, and also general set theory and transfinite arithmetic. He published mainly in mathematical journals, and also in a few philosophical ones. In 1899 Cantor suffered a severe mental breakdown which largely terminated his researches. However, in sad irony, just at that time, the opposition to his work, which had previously been marked in some quarters, was supplemented by a widespead acceptance of its basic legitimacy and importance. The principal aspects of philosophical interest are as follows. First, Cantor advocated that set theory could found integers and arithmetic, and thereby serve as a basis for mathematics. His means of procedure by idealistic processes of abstraction from sets to ‘define’ integers met little favour; but the programme influenced others, most notably the logicism of Bertrand Russell. Second, Cantor found criteria by which finite sets could be put into one-one correspondence, and especially when this was not possible. He also made clear the importance of the different ways in which members of an infinite set could be ordered. Third, Cantor’s study of the properties of transfinite numbers was based on an assumption, known as the ‘well-ordering principle’, which claimed that the members of every set could be laid out in that particular kind of order. In 1904 Ernst Zermelo showed that the proof of this assumption required the axiom of choice; nonconstructive character led to an intensive debate among mathematicians and philosophers. In addition. Cantor's ‘continuum hypothesis’, concerning the order of infinitude of the set of real numbers, led not only to technical results but enriched understanding of set theory and its models. Fourth, Cantor found that the usual definition °f continuity was necessary but not sufficient for the purpose. His search for additional defining properties led to Profound understanding of (discontinuities. Finally, Cantor was one of the first to find Paradoxes in set theory. For him the solution was simple: one should not advance as far as the absolutely infinite, that realm of sets large enough l° contain ‘everything’. When Russell found more Paradoxes in the 1900s, the status of paradoxes in set theory was raised, and a wide range of solutions sought which has influenced set theory, logics and philosophy ever since. Being is the fundamental question of philosophy but is not something which exists as an object set apart from the mind but is that upon which mind depends, the task of the mind being to come, by a never-ending critical reflection, to a proper awareness of the inexhaustible Being which is its foundation. The understanding that is sought will not emerge as a relation between understanding and a separate absolute objectivity such as God, for the objectivity that is sought is the foundation of subjectivity and not an object to be pursued by it. God is not one thing among many. For if God were an existent, God would always be another, whereas God is an intrinsic foundation. Being is composed of the plurality of existent subjects who seek it and constitutes a foundation for those seekers. That plurality of selves believes in a common world, but this is a matter of faith in an absolute which is the ground of our likeness to others. Knowledge is not knowledge of Being but knowledge-in-Being. Human thought takes place in this ambience of Being and is always the attempt to reflect upon its foundation in Being.