Although Russell's paradox has the virtue of simplicity, is it a distraction from other paradoxes of naive set theory?

Question

Given that Russell's paradox exhibits a contradiction in naive set theory, the interpretation of the binary relation "∈" called "membership" (where the expression "x ∈ m" is pronounced as "x is an element of m") is irrelevant. We could think of m as being like a bag of stuff and interpret ...

"x ∈ m" to mean that x isn't in that bag.

... and the same contradiction would arise regardless of the new way of thinking about the meaning of the primitive (i.e. foundational, rather than defined in terms of foundational concepts) concept of membership.

The usual explanation of why Russell's paradox arises is designed to motivate a particular approach to set theory, an approach that is hoped will not be discovered to be logically inconsistent, either in the form of ZF, or when additional axioms have been introduced and accepted in order to deduce hoped-for theorems such as Riemann's conjecture.

The usual explanation is that you simply cannot have a bag or "set" that contains every set x satisfying (x ∉ x), because that would be too much stuff in one bag or set. By invoking a connection between "bigness" and "infinity" and the ancient taboos against infinity, the explanation seems to psychologically satisfy some students of mathematics.

However, given that naive set theory is logically inconsistent, it remains inconsistent regardless of the interpretation. It's inconsistent when we interpret ...

"x ∈ m" to mean that x isn't in that bag.

So, clearly, the problem isn't that we cannot have a bag or set that has as elements some unacceptably large infinity of elements.

What are the essential bare minimum demands beyond consistency that a system of set theory should fulfill? It seems that those bare minimum demands should be discussed, and that there has been a process of looking for premises that will allow those bare minimum demands to be fulfilled. The axioms (such as for ZF) aren't self-evident. They are in a sense rationalizations, designed to reconstruct theorems already believed to be true prior to the invention of the premises.

Ordinarily, we proceed via premises with the understanding that any defect in our premises may call into question the truth of our conclusions. So, if we are going to build almost a whole subject area upon some one system of premises, and generally proceed as though there is no point in considering alternatives until after some consensus has been obtained in favor of the alternatives, then we really ought to believe that the premises are true. Students of mathematics who see the premises as conjectures have no particular reason to be impressed by various deductions -- using deductive logic applied more carefully in mathematics than in any other subject area -- of conclusions from those premises. Students of mathematics will naturally see the conclusions as supported by something, but not as definitely true.

Russell's paradox historically arose as a counter-example to Frege's simple assertion regarding the existence of sets corresponding to properties. Typically, when a very clear counter-example is provided for an assertion that seemed to be obviously true, the counter-example is surprising, but not too surprising. The assertion is usually accepted because it conforms with some vague ideas, and a picture emerges based on a small number of examples. A counter-example simply tends to show that the examples people had in mind when they were persuaded to accept the assertion was a rather impoverished set of examples that doesn't have anywhere near the richness of possibility encountered among the mathematical possibilities.

Is it possible that focus on Russell's paradox to the exclusion of other paradoxes of naive set theory has contributed to both narrowing the focus and motivating people to accept a system of set theory primarily because there doesn't seem to be any way to deduce -- within that system -- the contradiction that arises in Russell's paradox?

Answer 1

It is worthwhile pointing out that there are set theories having a set of all sets, hence the issue with the non-existent set of all sets not containing themselves is not about "size". Russell's paradox comes from unrestricted comprehension, and only becomes related to "size" if we keep set-restricted comprehension around.

Answer 2

According to Tim Button the reason Russell's paradox is a problem in set theory is because set theory relies on classical first-order logic and one can express that paradox there.

First he considers the paradox from the perspective of naive set theory: (page 109)

In part II, we worked with a naïve set theory. But according to a very naïve conception, sets are just the extensions of predicates. This naïve thought would mandate the following principle:

Naïve Comprehension. {x : φ(x)} exists for any formula φ.

Tempting as this principle is, it is provably inconsistent.

He then shows, after proving There is no set R = {x : x ∉ x}, that one can reformulate Russell's paradox in other contexts:

It is worth emphasising that this two-line proof is a result of pure logic. The only axiom we used was Extensionality. And we can avoid even that axiom, just by stating the result as follows: there is no set whose members are exactly the non-self-membered sets. But, as Russell himself observed, exactly similar reasoning will lead you to conclude: no man shaves exactly the men who do not shave themselves. Or: no pug sniffs exactly the pugs which don’t sniff themselves. And so on. Schematically, the shape of the result is just:

¬∃x∀z(Rzx ↔ ¬Rzz).

And that’s just a theorem (scheme) of first-order logic.

He can then reach his conclusion that the problem is not with set theory but first-order logic:

Consequently, we can’t avoid Russell’s Paradox just by tinkering with our set theory; it arises before we even get to set theory. If we’re going to use (classical) first-order logic, we simply have to accept that there is no set R = {x : x ∉ x}. The upshot is this. If you want to accept Naïve Comprehension whilst avoiding inconsistency, you cannot just tinker with the set theory. Instead, you would have to overhaul your logic.

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Button, T. Set Theory: An Open Introduction. Retrieved on October 4, 2019 from the Open Logic Project at http://builds.openlogicproject.org/courses/set-theory/settheory-screen.pdf

Answer 3

To pile on Arno's answer, non-containability really has nothing to do with size. Having members of every size is a simple test, but it is not the basic reason any set is not containable. It is actually quite an indirect one. Getting from the one fact to the other formally involves sets having cardinalities, cardinals being ordinals and the Burali-Forti paradox.

The set of all integers not definable in under sixty letters is also not containable, even though it clearly has in it only positive integers. But if it were a set, of positive integers it would have a least element, and precipitate the Berry Paradox.

(In symbols, say we define the 'un-quoting' operation analogous to the Linux backtick in the language of our set-theory. Let `s` be the opposite of quoting, in that it takes a string s and returns the interpretation of that string in the ordinary language of set theory and arithmetic (fully complemented with string operations, infimum and supremum operators, etc.)

Now let define the string B = "{x:∀t`t`=x→|t|>25}", (where || takes the length of a string). So B is the set of things we can define in no less than 25 symbols of our language. Then `B` can't be a set. If it contained no natural numbers, then any number we could define in 25 or more characters, we could always define in fewer, and we would need no more than 25 characters to make any arithmetic definition. That is clearly not true. So it contains a natural number, and therefore a least one. But If we choose b to be the least natural number in `B` and let t = "⋀" + B. Then `t` = b, and b is in `B`, but |t|=19.)

The size metaphor clearly no longer works in set theory with a Universal Set like Goedel-Bernays-von Neumann. It is a total red-herring used as an excuse, that does not actually mean anything.

So yes, every historical paradox can probably be manipulated into generating naive sets that are not ZFC sets. The set of all integers that would enumerate heaps of sand. The set of all statements that can be made by liars. Etc. There is more wrong with our intuition of assignment and definition than just forbidding this single set, and avoiding infinities that are somehow 'too big'.

You can claim that notions like 'definable', the meanings of lies, and the vagueness of English words are not part of set theory. But there is no good reason to rule out their translation into those terms. They are still part of our overall naive notion of definition and therefore set comprehension. Russell's is just so simple that one can easily translate it into symbols.