Russell's paradox

Russell's paradox is a paradox that arises in a certain "naive" approach to set theory. The reasoning behind it is as follows.

We begin with the plausible-sounding premise that to any logical predicate $P(x)$ there exists a set $\{x:P(x)\}$. Indeed, there is some way in which predicates and sets seem like they are the same thing. One way I can think about this object $x$ on my table is to say that it is a banana, or $B(x)$; another way I can think about the same fact is to consider bananas as a set $\{y:B(y)\}$, and say that $x$ belongs to this set, or $x\in \{y:B(y)\}$. If it makes sense to say that $x$ is a banana, then it makes sense to consider the set of all bananas.

Now, let's consider the predicate $E(x):=(x\in x)$, which makes sense to state if $x$ is a set. Although it is difficult to think of examples of sets that contain themselves, this predicate is not as ridiculous as it might seem. For example, the set of triangles (if appropriately defined) is itself a triangle. Or, for example, "$x$ is a set" is a well-defined predicate, so therefore there exists a set of all sets -- and since it is a set, it must necessarily contain itself.

The paradox comes in when we consider the set $R=\{x:x\not \in x\}$, and ask: does $R$ contain itself, i.e. is $R\in R$? If the answer is yes, then $R\not \in R$, which would be a contradiction. If no, i.e. if $R\not \in R$, then $R\in R$, which is again a contradiction.

The solution

The essence of the following solution was explained to me by Harry Binswanger in his Philosophy of Mathematics course (2024), and in his book How We Know. I have added a few embellishments of my own.

The heart of the matter is that we have to be more careful about what we mean by a "set." Sets don't just exist "out there," but rather a set is some things out there, which someone is considering together as a single unit. Sets are concepts of consciousness, and so they are relational. They are things out there, as viewed in a particular way by a particular man. A set is an action of awareness, or of referring.

To refer is to refer to something. A reference to an act of referring can be fine in certain circumstances, e.g. I can unproblematically refer to the fact that I referred to Harry Binswanger's book How We Know. But an act of referring that refers to nothing but itself is a contradiction in terms. It would be referring to its referring to its referring to its referring to ... etc. We would get an infinite regress, and so this supposed act of reference would in fact be referring to nothing at all, i.e. it would not be an act of reference. Harry Binswanger calls this "the fallacy of pure self-reference."

Now, consider the implications of this logical fact for statements in set theory. Self-referential sets, like "the set of all sets" cannot exist. Why? Because to consider all sets together as a set is to refer to this particular act of reference. And what is this particular act of reference referring to? Well in part, it is referring to its own referring (and its own referring is referring in part to its own referring, to its own referring, to its own referring, to ...). Again, we get an infinite regress.

Implications

We have arrived at the surprising fact that the statement that "$x$ is a set" is perfectly unproblematic, but that "the set of all sets" is not something which can be rationally considered. Usually, to form a set, it is sufficient just to know what you are referring to.

An implication of this is that the concept "set" is unlike most other concepts, in that its referents cannot be considered together as a single unity, for one of its referents would be that act of consideration itself. This must have implications for generalizations about sets, but it's hard to say what.

Russell's paradox does have some implications for "set formation rules." Traditionally, the way it is dealt with is by only allowing sets to be formed that are subsets of some other larger set, and then postulating the existence of a universe (a very large set, that everything else can be a subset of).

Traditional math takes an intrinsic approach to constructing the universal set, but I take an objective approach to it. The "universal" set is: existence, as it is known by a particular consciousness. It is the set of all the things that are known to exist by a particular consciousness. That is, in order for you to validly form a set, it must consist exclusively of things that exist, and which you are aware of. Each individual has his own "universe," from which he is allowed to form sets; the universe is different for different individuals, and the scope of an individual's universe changes with his knowledge.

If, in forming sets, you only ever include things in your own "universal" set, then you cannot possibly commit the fallacy of pure self-reference. Your acts of awareness exist, but there are limits to how aware of them you can be.