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.

Answer 2

I'm definitely not an expert in set theory, but Russel's paradox has long since been dealt with by making the class–set distinction.

$V$ is the class of all sets, not the class of all classes or the set of all sets, and this is really the whole shebang. We aren't allowed to collect 'all collections of the same nature' (sets or classes) into a collection of that same nature (one big set or class) on pain of paradox, but we can collect all collections of a certain nature (sets) into a new, bigger type of collection with a different nature (a class).

We can even continue this hierarchy with 'hyperclasses' that are allowed to hold all classes but not other hyperclasses, etc, as explained in the answer to An axiom for collecting proper classes by Joel Hamkins (and Andreas Blass/Kameryn Williams in the comments on Joel's answer).

In essence we can allow for a fundamentally 'bigger' type of collection, which can then hold all collections of a smaller type, but this new bigger type of collection will still never be able to collect up all collections of its own type — we would have to once again step higher up the 'collection hierarchy', at which point we would run into the same situation again.

Answer 3

Russell's paradox depends on the local admissibility of universal exclusion, i.e. the "all and only" quantifier. Consider the difference between:

1. $\forall a\exists X((a \notin a) \implies (a \notin X))$
2. $\forall a \exists X((a \in X) \implies (a \notin a))$.

To get the Russell set, you have to merge (1) and (2) by composing the conditional arrow into the biconditional arrow, which is where the "and only" part of the universal exclusifier comes in.

The "upshot" is that you can deny (2) while retaining (1): you can say that there is some set that happens to have all noncircular elements to its name. However, if these are the only elements that set contains, then Russell's paradox arises, so we would have to say that if there is a set with all noncircular sets as elements, this set still is an element of itself on the side. This whole set does not have its elements just because they've been intensionally carved out as such, but either the whole set is "manually" self-inserted, or it satisfies a disjunctive property like, "All elements of $X$ are either noncircular or are $X$ itself."

Note, then, that there are set theories with universal sets, e.g. Quine's parafounded world, or Weber's paraconsistent $\aleph_\text{ORD}$. At least in the consistency-friendly such cases, if there is a universal set, then the axiom scheme of separation has to fail for this set, the powerset axiom to boot: for a truly universal set, there is no difference between its elements and its subsets, or the concept of it having subsets is not well-thought, so there are no separable subsets or separate sets of subsets in play, here.

Given phenomena like the conflict between zero sharp and the constructible universe, or the Kunen inconsistency, it is possible to reformulate most or perhaps all axioms as dependent in the sense that they "run out" at some point in the cumulative hierarchy. For example, you can compromise on replacement, foundation, or usually general choice, to work around the Kunen inconsistency (Corazza's approach gives us Reinhardt cardinals below a certain iterate of the embedding function; antifoundation worlds have n.e.e.'s involving their parafounded fragments but can retain the Kunen wall for their well-founded fragments). These phenomena then support what we might call "deflection arguments," which contrast a universal set with relatively infinite sets and lend themselves to an anticlass principle: "If a set contains all of a certain kind of element, then it contains all elements of all kinds whatsoever." E.g., if there is a set of all ordinals, there is yet no set of all and only ordinals, but this set has to have all the cardinals in it, too, etc.

Yet another solution (from the separation-minus vantage, but comporting more strictly with the axiom of foundation) would be to invoke couniversal sets. The pure example would be $$\exists X \forall ab((a \neq X) \iff (a \in X)).$$

The propriety of calling these "couniversal" is that they can be styled "sets of all other sets." In a world with a universal set simpliciter, then the deviant consequence obtains wherein the couniversal $X$ contains its superior as an element, which also contains $X$ in turn, so $X$ ends up being cofounded. So to avoid this, one would have to go to a world where there's a couniversal set without a universal set above it. In this event, $X$ is the set of all elements, not all sets (since $X$ itself is not, in that world, an element of any set, not even itself).

So then a couniversally noncircular set would be "the set of all other noncircular sets," which indicates its own noncircularity.

All these solutions end up requiring a stronger distinction between extensional and intensional elementhood parameters. Cantor infamously mistook definability for orderability, and in fact per the natural language in use at the time, Cantor's system involved a partial collapse of the extension-intension distinction into the cardinal-ordinal one. We "know better," though, so we can talk about an elementhood parameter that is "manually extensional" rather than a matter of elements satisfying intensions. (I wish I could find it again, but I read an essay in which they explicitly brought up a difference between "is an extensional element of" and "is an intensional element of," which I saw as a way to expand on double-extension set theory, which itself involves a relatively novel attempt to deal with Russell's paradox.)