Academic journal article The Review of Metaphysics

Cantor's Concept of Set in the Light of Plato's Philebus

Academic journal article The Review of Metaphysics

Cantor's Concept of Set in the Light of Plato's Philebus

Article excerpt

IN THE OPENING SENTENCE of his final mathematical publication, Beitrage zur Begrundung der transfiniten Mengenlehre, Cantor states in a few words what h.e means by a set.

   By a "set" we understand every collection M of definite
   well-distinguished objects m of our intuition or our thinking
   (called the "elements" of M) into a whole.'

Here, however, we shall focus on an earlier "definition" that intimates concerns about the ontological status of collections. It appeared in Cantor's Grundlagen einer allgemeinen Mannichfaltigkeitslehre (2) which summed up the quintessence of his deepest contribution to mathematics--the transfinite numbers--both from a mathematical and a philosophical point of view. In an end note Cantor emphasizes that Mannichfaltigkeitslehre is to be understood in a much more encompassing sense than that of the theory of sets of numbers and sets of points he had been developing up to that stage.

   For by a "manifold" or "set" I understand in general every Many
   which may be thought of as a One, i.e., every totality [Inbegriff]
   of determinate elements that can be united by a law into a whole.

He then goes on to explicate this further by drawing upon Plato's Philebus.

   And with this I believe to define something related to the Platonic
   eidos or idea, as well as that which Plato in his dialogue
   "Philebus or the Supreme Good" calls mikton. He contraposits this
   against the apeiron, i.e., the unlimited, indeterminate which I
   call inauthentic-infinite, as well as against the peras, i.e., the
   limit, and he declares the [mikton] an orderly "mixture" of the
   latter two. That these two notions are of Pythagorean origin is
   indicated by Plato himself; cf. A. Boeckh, Philolaos des
   Pythagoreers Lehren. Berlin 1819. (3)

The aim of this article is to shed some light on these two passages in Grundlagen. Specifically, I propose a substantive connection between Plato's mature theory of ideas and an interpretation of Cantor's 1883 definition that, arguably, is in accordance with the meaning of "set" in contemporary set theory. Among other things this will allow a construal of the term "law" that does not entail any restriction to definable sets.

For the most part, the perspective from which Cantor's thought will be presented here is that of Grundlagen. In particular, this means that the ordinals are conceived of as numbers "constructed from below" by the principles of generation described in [section] 11 of that work in contrast to the "purely mathematical" treatment of ordinals as canonical representatives of order types of well-ordered sets adopted by Cantor in all of his subsequent publications. (4) I also incorporate notions which, although they found their explicit expression at later stages, were undoubtedly known to the author of Grundlagen. (5) Similarly, ideas from contemporary set theory such as large cardinals are discussed. They are not explicitly present in Cantor's writings but nevertheless can be traced back to the fundamental intuitions underlying Cantorian set theory. The question in what sense Cantor's 1883 definition differs from the one of 1895 as well as other aspects of the latter will be investigated on another occasion.

Our interpretation of Plato relies, to the extent possible, on the systematic exposition by Eduard Zeller. (6) The intention, however, is not to vindicate this author against numerous studies that have been undertaken since then, but to use a source that Cantor evidently regarded as congenial to his own reading of Plato. (7)

I

Set, Eidos and Idea. On ontological grounds, Cantor's comparison of sets with the Platonic eidos or idea is thoroughly intelligible. Either one of those notions may be conceived of as a One over Many insofar as they denote what is common to a plurality of individuals of the same name. (8) For Plato, this One over Many is an autonomous entity just as, for Cantor, a set is distinct from the multiplicity of its elements. …

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