Why can't Russell's Paradox be solved with references to sets instead of containment?
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.
Let me present a different perspective. Instead of modeling a set using a container object consisting of pointers/references, which can be limiting, suppose we choose to model sets as a function which
- will accept any set, and
- always returns a boolean.
How this is to be implemented in any specific language, such as Java, isn't really relevant. There are structural issues to overcome; Java presents the difficulty that a function is not an object, for example, and my code does not address this issue. This is, however, not a serious hurdle for the concept.
The important thing is we can certainly implement the universal set:
bool Universe(Object X) { return true; }
and we can implement what ostensibly should be the set of all things which do not contain themselves:
bool Russell(Object X) { return !X(X); }
So what's wrong with this set Russell? Well, if you plug Russell into itself, then it will call itself, creating an infinite recursive loop. Therefore, Russell fails to return anything when plugged into itself as a parameter, so it cannot be a set.