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Cantor's general theory of sets

It was a widespread idea in the late nineteenth century, that pure mathematics was nothing but an elaborate form of arithmetic and, further, that the “arithmetisation” of mathematics had brought about higher standards of rigor… This viewpoint led to the idea of grounding all of pure mathematics in logic and set theory, which proceeded in two steps:

1. the establishment of a theory of the real numbers (arithmetization of analysis)
2. the set-theoretic definition of the natural numbers and axiomatization of arithmetic

Infinite sets had been needed for an adequate definition of important mathematical notions, such as limit and irrational numbers. It was this initial use that led Cantor himself to begin studying infinite sets “in their own right.”

In 1883, Cantor began his general theory of sets with the publication of Foundations of a General Theory of Manifolds (the Grundlagen). Among other things, in this paper he made the interesting and somewhat self-justifying claim of the autonomy of pure mathematics:

• pure mathematics may be concerned with systems of objects which have no known relation to empirical phenomena at all
• any concepts may be introduced subject only to the condition that they are free of contradiction and defined in terms of previously accepted concepts

Cantor’s overall purpose, however, was to develop a theory of transfinite numbers. The significance of Cantor’s general theory of sets not only for the process of mathematical rigourization generally, but also for what Hilbert would later state as his 2nd problem particularly, was this:

precisely when mathematicians were celebrating that “full rigor” had been finally attained, serious problems emerged for the foundations of set theory.

Neither the initial introduction of infinite sets by others nor their use in his early theory of sets by Cantor himself had been problematic. However, Cantor’s subsequent introduction of transfinite numbers and development of transfinite arithmetic made him aware of the potential for paradoxes within set theory.

Cantor is said to have attributed the source of these paradoxes to the following:

• the use (by Frege) of an unrestricted principle of comprehension
• the acceptance (by Dedekind) of arbitrary subsets of a Universal Set (Gedankenwelt)

More specifically, the claim is that Cantor himself traced the paradoxes to a faulty understanding (by others) of what constitutes a legitimate mathematical collection. For Cantor, the mathematically relevant notion of a collection is said to have been based on the “combinatorial concept” of a set:

In order to be treated as a whole, [a mathematical collection] must be capable of being counted, in a broad sense of "count" which means well-orderable.

In contrast to this was the “logical concept” of a set, developed by Frege, accepted by Dedekind, and championed by Russell, which “treats collections as the extensIons of concepts”:

For a multiplicity to be treated as a mathematical whole, we must have some propositional function which acts as a rule for picking out all of the members.

The point of this contrast rests on the claim that the set-theoretic paradoxes are only a problem for the logical concept of a set, which includes the inconsistent Comprehension Principle, and has Russell's Paradox as a result.

The general consensus, however, is that it was Cantor’s own construction of the system of transfinite numbers, employing as it did the concept class of numbers, that introduced foundational problems into mathematics by asking this question:

What cardinal numbers are there?

In the Grundlagen, there was a significant change in Cantor’s conception of a set, which he defined as follows:

any multiplicity which can be thought of as one, i.e. any aggregate (inbegriff) of determinate elements that can be united into a whole by some law.

With the introduction of the transfinite numbers, Cantor immediately recognized that this notion of set was problematic:

not every property of numbers “unites the objects possessing it into a whole”

The most notable changes in this from his earlier explanation of the concept of set are these:

• the absence of any reference to a prior conceptual sphere or domain from which the elements of the set are drawn
• the modification according to which the property or “law” which determines elementhood in the set “unites them into a whole”

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References

• Curtis, G N. (1995). “The ‘Villain’ of Set Theory,” [Review of Lavine, S. (1994). Understanding the Infinite, Harvard Univ. Press], Russell: The Journal of Bertrand Russell Studies, Vol. 15, No. 1, 1995, McMaster University, URL: https://escarpmentpress.org/russelljournal/article/viewFile/1882/1908, Accessed: 2015/07/16.