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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.

The “serious problems” that emerged were, of course, the set-theoretic paradoxes.

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, his 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, both then and now, however, is that Cantor’s own construction of the system of transfinite numbers introduced foundational problems into mathematics by asking (and then answering!) this question:

What 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.

The 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”

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”

Further, in introducing the transfinite numbers, Cantor employed the notion of set in an entirely new way:

• Cantor’s previous notion of a set involved specifying a set of objects from some given domain, albeit one which was already well-defined.
• Cantor’s new notion of a set introduced the transfinite numbers in terms of the notion of a set of objects of that very same domain.

Cantor defined numbers to be what can be obtained, by starting with the initial number ($0$) and applying two operations, which he called principles of generation:

1. the usual process of taking successors, which yields, for every given number $\alpha$, its successor $\alpha + 1$
2. a new process of taking limits of increasing sequences, which yields, after any given sequence of numbers without a last element, a number $\beta$

Cantor intention was "to generalize in a rigorous way the very notion of number in itself ... by building transfinite and finite numbers, using the same principles."

Cantor defined numbers (including the transfinite numbers) in terms of the notion of a set of numbers, as follows:

Let $Ω$ denote the class of all numbers

and $X$ range over sets

and $S(X)$ be the least number greater than every number in $X$, such that

if $X$ is a unit set of element $\alpha$, i.e. $\{\alpha\}$

then $S(X)$ is (as usual) $\{\alpha+1\}$

but if $X$ is a set (sequence) of numbers with no greatest element,

then $S(X)$ is the limit of the sequence of numbers in $X$ arranged in natural order

and $X \text{ is a subset of } Ω ⇒ S(X) ∈ Ω$.

Then we have

• $S(∅)$ is the least number $0$
• $S(\{\alpha\})$ is the successor of $\alpha$, i.e. $\{\alpha + 1\}$ OR $S(X)$ is the limit of the sequence of numbers in $X$ arranged in natural order

Further,

• $Ω$ is well-ordered by $<$ (there are no infinite descending sequences of $\alpha_n$).

There are disagreements concerning Cantor’s precise notion of number and how that notion related to the concept of a well-ordered set. There is also disagreement concerning the paradoxes of which he actually aware. There is, on the other hand, general agreement that Cantor understood the paradoxes to be “a fatal blow to the ‘logical’ approaches to sets favoured by Frege and Dedekind” and that, as a result, he attempted to put forth views that were opposed to the “naïve assumption that all well-defined collections, or systems, are also ‘consistent systems’.”

The paradoxes convinced both Hilbert and Dedekind that there were important doubts concerning the foundations of set theory. Cantor apparently planned to discuss the paradoxes and the problem of well-ordering in a paper that he never actually published, but the contents of which he discussed in correspondence with Dedekind and Hilbert.

A partial summary of Cantor’s achievements that arose from his general theory of sets includes an “almost modern” exposition of the theory of well-ordered sets and also the theory of cardinal numbers and ordinal numbers. As he introduced the terms, an ordinal number was the order type of a well-ordered set and a cardinal number was the equipollence type of an abstract set.

In the view of some (but certainly not all) mathematicians, Cantor’s study of infinite sets and transfinite numbers introduced little that was alien to a “natural foundation” for mathematics, which “would, after all … need to talk about sets of real numbers” and “should be able to cope with one-to-one correspondences and well-orderings.”

Here is a succinct and remarkably robust defence of what Hilbert subsequently called “Cantor’s paradise”:

There are many mathematicians who will accept the ... theory of functions as developed in the 19th century, but will, if not reject, at least put aside the theory of transfinite numbers, on the grounds that it is not needed for analysis. Of course, on such grounds, one might also ask what analysis is needed for; and if the answer is basic physics, one might then ask what that is needed for. When it comes down to putting food in one’s mouth, the ‘need’ for any real mathematics becomes somewhat tenuous. Cantor started us on an intellectual journey. One can peel off at any point; but no one should make a virtue of doing so.

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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.

• Nunez, R E. (2003). “Creating mathematical infinities: Metaphor, blending, and the beauty of transfinite cardinals,” Journal of Pragmatics, Vol. 37, (2005), pp. 1717–1741, URL: http://www.cogsci.ucsd.edu/~nunez/web/TransfinitePrgmtcs.pdf, Accessed: 2015/010/02.