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What is the principal difference between Russell paradox and any negative proof?

In Russell paradox we are suggesting the existence of the set of all sets not containing themselves and coming to contradiction. This just means that the set doesn't exist and nothing more.

Why we say that this means something bad about naive set theory?

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up vote 6 down vote accepted

Naive set theory was (for the most part) centred around the notion of Unrestricted Comprehension: Given any property $\mathcal{P}$ there exists a set whose elements are precisely the objects $x$ satisfying property $\mathcal{P}$. A perfectly good mathematical property would be "$x$ is a set and is not an element of itself," and so there must be a set $R$ whose elements are exactly those sets which are not elements of themselves. This is Russell's paradoxical set.

So $R$ proved that Unrestricted Comprehension was inconsistent, and we thus have to disallow the construction of this "set" $R$, but still allow for the construction of interesting sets. We could, of course, simply say that Unrestriction Comprehension is fine, except for this set. However you then can still arrive at the Burali-Forti paradox. So maybe we also disallow the the construction of this problematic set. But there will be yet more problematic sets. At any rate, it would seem to be impossible to prove that by simply disallowing the construction of sets appearing in some list of known "paradoxical properties" that the resulting theory becomes consistent.

Much better would be to describe principles of set construction which would seem to disallow the construction of these paradoxical sets. The most common axiomatization now, ZFC, has the following result:

Fact: If ZFC is consistent, then there is no set of all sets. (This is true even for a very small fragment of ZFC.)

Proof: If there were a set of all sets, call it $V$, then by the Axiom Schema of Separation we can construct the set $W = \{ x \in V : x \notin x \}$. However as $W$ is a set, then $W \in V$, and it follows that $W \in W$ iff $W \notin W$! $\Box$

Unfortunately, due to certain amazing theorems of Kurt Gödel, we cannot prove that ZFC is consistent (without transcending ZFC itself) and so there is no proof that ZFC does in fact avoid all paradoxical sets. But several decades of intense research has not yielded any contradictions, so it is fair to say that most set-theorists believe in the consistency of ZFC.

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Why can't we consider EACH paradoxial set as just not existing? Turn to number domain, to the prime numbers. Nobody knows the law P(n) for them. So what? Should we reinvent arithmetic where prime numbers are predictable? No. Arithmetic remains as is. So as should be in set theory. Yes, there are some sets which are not exist and we cannot state simple law to detect them. We can prove this only with unique proofs for each set. So let it be. Let these "paradoxial" sets be as prime numbers -- strangely distributed among others. – Dims Jan 24 '13 at 12:43
@Dims: I believe that the main with this approach is that the resulting theory would be so complicated that even in principle a computer could not be programmed to distinguish between valid instances of Unrestricted Comprehension and "paradoxical" ones. Such a theory would have the draw-back that it would not be possible in general to tell good proofs from bad ones. (cont...) – arjafi Jan 24 '13 at 12:52
(...inued) Compare this to Gödel's Incompleteness Theorems, which states that there is no recursive, complete, consistent theory of arithmetic; of course there is a nonrecursive, complete, consistent theory of arithmetic: namely all statements true of the structure $( \mathbb{N} , 0 , 1 , + , \cdot)$. But this is so complicated that one cannot tell the axioms from the non-axioms. – arjafi Jan 24 '13 at 12:56

In naive set theory, we claim that it is possible to construct sets by saying which elements it contains and which not (all elements from universum have to satisfy some sort of rule or predicate to be in set). And the definition of Russel's set is completly right in naive set theory, because it says clearly, which elements it contains and which not. Since it's well defined in naive set theory, but leads to a contradiction, we obtain a paradox.

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Within naive set theory, the Russell paradox is a complete positive proof of the absurd. There is nothing negative about the proof (like assuming the contrary of what is to be proved), but if you want to prover the absurd, you do so by arriving at a contradiction; this is the essence of the absurd. Having a proof of the absurd, the theory one is working in (here naive set theory) is clearly inconsistent, and you could in fact prove anything that you can formulate in it.

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