[Math] Russell’s paradox as understood by current set theorists

Question

Many mathematicians like to think of the set of natural numbers as existing as a completed object. But it is difficult to make set theory as concrete, because Russell's paradox, in conjunction with some type of separation principle, tells us that arbitrary "collections" cannot be collected into a completed object. I view this as telling us that the metaphysical idea of "collection" has some natural limitations that we might not have realized, a priori.

Now, in terms of the formal mathematics of collections—known as set theory—there seem to be two standard fixes to address the paradox.

Class and set distinction First is the idea of creating a new level of collection called "proper classes". In some set theories like ZFC, classes are an informal notion referring to the formulas of the language. Some mathematicians still view those classes as referring to meta-collections in the metatheory. They even use set-builder notation to refer to them. In other versions of set theory, like NBG or KM, classes are also formal objects. Sometimes they are of a different type than sets, and sometimes sets are classes with extra properties.

Those theories with classes can often be reinterpreted inside the theories without classes, and vice versa. Thus, it seems that Russell's paradox does not prescribe the existence, Platonically speaking, of two distinct types of collections—the set and the proper class. Yet this language has also become very useful to mathematicians. My question is somewhat philosophical in nature. Do modern set theories view proper classes as a necessary, true concept? Do they favor the view that proper classes are only informal, or are they formal?

I have a follow up question, for those set theorists that believe a "true Platonic set theory" exists. How do you view that completed set theory in light of Russell's paradox? It seems that a "true set theory" couldn't be like a collection itself (hence not like a set, nor like a proper class even). In particular, "true Platonic set theory" would be unlike any model of formal set theory, since the domain of a model is a collection.

Type theory Another solution, which I am much less familiar with, is using type theory to limit collection principles. Are there many modern set theorists who favor this resolution? Or has the proper class idea overriden this solution?

Answer 1

Let me begin quoting W. Tait (lectures on proof theory, pages 4 and 5):

I believe that what further has to be understood, in order to make sense of these 'paradoxes' is that the notion of a transfinite number or, equivalently, of a set of transfinite numbers is an essentially open-ended notion: no matter what principles we introduce to construct sets of numbers, providing only that these principles are well-defined, we should be able to admit all numbers obtained by these principles as forming a set, and then proceed on to construct new numbers. So $\Omega$ cannot be regarded as a well-defined extension: we can only reason about it intensionally, in terms of those principles for constructing numbers that we have already admitted, leaving open in our reasoning the possibility - in fact, the necessity - of always new principles for constructing numbers. When this is not understood and $\Omega$ is counted as a domain in the sense of a well-defined extension, then the so-called paradoxes force on us a partitioning of well-defined extensions into two categories: sets and proper classes; and the only explanation of why such an extension should be a proper class rather than a set would seem to be simply that the assumption that it is a set leads to contradiction. The paradoxes deserve the name 'paradox' on this view because it offers no explanation of why there should be this dichotomy of well-defined extensions.

Emphases are mine ($\Omega$ is a reference to "the greatest ordinal"). So, one proposed rough sketch of an answer in the direction given by Tait (of course, there are many other directions in philosophy) is this:

• The subject matter of set theory is open-ended, therefore set theory must be about an intension, the concept of set, not about a well-defined extension. This intension is open-ended (it is hard to make sense of the oxymoron "open-ended well-defined extension"), and it is the unifying criterion behind the plurality of set theoretical practices. The contemporary criterion can be more or less approximated by $ZFC$, but there can be no definite final stage on the progressive conceptual unification of the set-theoretical practices, as there is a neccessary open-endedness (incompleteness) in this intension.

There are many things to address here, but I will not try to pursue them, not even in outline, as this would lead us to a more hardcore philosophical activity. As a final remark, there are similar arguments in the history of philosophy which were given many years before Russell. One of the most relevant is Plato's third man argument, in Parmenides.

SPECULATIVE ADDENDA:

I think the question "should there be a dichotomy of well-defined extensions and how can we deal with it?", a natural outcome of this discussion, is very relevant for the foundations of set theory, and there are many hints about this in traditional philosophy, say, from Plato to Hegel. I think the answer is no, and I agree with Tait's direction. (A small digression: "Platonism", as the term appears in the original question, has probably a very weak connection to Plato. Plato is very subtle, he wrote dialogues, not theoretical treatises in philosophy, in which the dramatic elements interact with the philosophical elements, probably because he sees philosophy as the kind of argumentative activity he shows in the dialogues, not as a body of theory. See W. Tait, Truth an Proof: The Platonism of Mathematics. Anyway, I think, along with Tait, that the man deserves a better fate.)

I will not dare to say much more about our questions here, but I would like to remark on the idea that there can be no final conceptual unification, for any unification is synthetic, that is, made on the basis of a new conceptual synthesis which, as "new", cannot be among those very things now unified. If reason operates this way, progressively unifying its previous practices through conceptual synthesis, open-endendness is its fate, and I believe mathematics is the primary example of this.