[Math] Why can’t Russell’s Paradox be solved with references to sets instead of containment

Question

My background is in computer science, and I'm keeping the Java implementation in my mind as a model. Included in the Java language is the notion of sets.

Now I understand that this is different from the model Russell and Whitehead had in their minds when they were writing Principia Mathematica, but I don't completely understand why it is different.

To me, when you say "a set that contains a set," you have three ways you can "implement" this. You can say that it is "physically inside" (and draw it inside). You can say "it is just a logical concept" (which is what I think Russell was getting at). And you can say "it is a physical concept, but not physically inside — we link them together with pointers" (like in computer programming).

Taking this further into Russell's paradox: "The set of all sets that don't contain themselves," when talking about computer programming, is a relatively easy concept to implement (within the domain of the sets in a computer program).

I'm guessing there is a philosophical difference between sets in Java and Russell's sets. (I imagine there must be a name for Russell's sets, but I don't know what they are called.)

I can see that mathematics has other theories of sets like Zermelo–Fraenkel set theory and Quine's New Foundations.

My question is: Why can't Russell's Paradox be solved with references to sets instead of containment?

Answer 1

On the contrary, Russell's set - interpreted in your computer-programming model - is not easily implemented (let alone possible). Actual "containment" is not an issue - in principle, mathematicians would be perfectly happy with a set that contained itself.

The issue is this: consider a (Java) Set that contains in it references to exactly those Sets currently in memory which do not contain references to themselves. Call this Set<Set> R. Suppose R contains a reference to itself. Then R does not meet the requirement for being referenced by a member of R, so R cannot be referenced by anything in R, contradicting our supposition. So suppose instead that R does not contain a reference to itself. Then R meets the requirement for being referenced in R (that is, it is a Set which does not contain a reference to itself) so it must have a reference in R, contradicting our supposition.

Again, whether or not containment actually means "containment" isn't relevant. In fact, modern set theory formally treats membership abstractly - the symbol $\in$ doesn't have any canonical meaning, it's just an arbitrary relationship that obeys certain axioms. It's helpful to visualize it as actual containment or as a system of Java-like references, because those visualizations obey the axioms, but the "physical" implementation isn't relevant to any of the logic.