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It is well known that something goes wrong with Russell's class of all classes which are not members of themselves. If this class is a member of itself then it is not a member of itself, and if this class is not a member of itself then it is a member of itself. Contradiction!

Nevertheless, it is generally believed that nothing goes wrong with the decent class of all classes which are members of themselves. Namely, if this class is a member of itself then it is a member of itself, and if this class is not a member of itself then it is not a member of itself; and nothing is wrong with that.

But we could ask whether Russell’s class of all classes which are not members of themselves belongs to the decent class of all class which are members of themselves. If Russell’s class belongs to the decent class, then Russell’s class belongs to Russell’s class and then Russell’s class does not belong to Russell’s class. Hence Russell’s class does not belong to the decent class. On the other hand, if Russell’s class does not belong to the decent class, then Russell’s class does not belong to Russell’s class and then Russell’s class belongs to Russell’s class. Hence Russell’s class still belongs to the decent class. Contradiction!

We can conclude that something goes wrong with the decent class too. Perhaps we should blame the member, not the class. But I think that paradoxical members simply do not belong to non-paradoxical classes, do you?

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    "something goes wrong with Russell's class of all classes which are not members of themselves" Correct; thus, this class does not exist. So we cannot "ask whether Russell’s class of all classes which are not members of themselves belongs to..." – Mauro ALLEGRANZA Oct 3 at 12:17
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    What does "decent" mean as applied to a class? And "something" goes wrong already with the class of all classes, although it is not inconsistency. While the truth value of membership in the Russell's set is overdetermined (hence contradiction), it is underdetermined for the class of all classes. This is analogous to the liar and truth teller paradoxes, respectively – Conifold Oct 3 at 21:45
  • Mauro ALLEGRANZA, You should think in the context oh Quine's NF, cf. my Cantor’s theorem and paradoxical classes, Zeitschrift für math. Logik und Grundlagen der Math., Band 32, pp. 221-6, 1986. (fsb.unizg.hr/matematika/sikic/download/ZS_cantors_theorem.pdf) – Zvonimir Sikic Oct 4 at 8:55
  • Conifold, Decent class is just the name of the introduced class. It can be formally defined in Quine's NF (cf. comment above) but I presented it informally , so as to be understandable to wider audience. – Zvonimir Sikic Oct 4 at 9:01
  • @Conifold: I suspect a translation issue of some kind. They may mean a "proper class" (i.e. a class which is not a set)? – Kevin Oct 5 at 17:15
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It is well known that something goes wrong with Russell's class of all classes which are not members of themselves.

Wait a moment. Russell's paradox was about sets, not classes. You cannot simply replace the word "set" with the word "class" and hope to recover the original paradox in modern (axiomatic) set theory.

Let's define some terms:

  • A class is a collection of objects which satisfy some predicate of first-order logic. In general, classes are not necessarily members of the domain of discourse, and (therefore) cannot necessarily be members of other classes. Therefore, you can't reconstruct Russell's paradox out of classes.
  • A set is a class which additionally satisfies the axioms of (usually) Zermelo-Fraenkel set theory with the axiom of choice (ZFC for short). Because sets satisfy the axioms, they are members of the domain of discourse and may appear as members of other classes, including other sets. The axiom of regularity (in ZFC) forbids a set from directly or indirectly containing itself. Set theories which lack an equivalent axiom are called "non-well founded set theories," and are not paradoxical or otherwise problematic, just different.

The object which your sentence (quoted above) describes does not exist. The closest equivalent (under ZFC) is the class of all sets. For non-well founded set theories such as Quine's New Foundations (NF), then you are discussing the class of all sets which do not contain themselves. This class is not itself a set, because its formula is not stratified (which is required by NF's axiom schema of comprehension), so you still cannot reconstruct Russell's paradox in NF.

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  • Logicians started with unrestricted comprehension. Russell discovered a paradoxical class and we had to give up unrestricted comprehension. Then we adopted restricted (limitation of size) comprehension, usually called separation, which excludes Russell class, my decent class, universal class etc. But some logicians were not so restrictive, e.g. Quine in NF adopted stratified comprehension which does not exclude the universal class (it is not paradoxical) but it excludes Russell's class as paradoxical. My question is paradoxality of the decent class (the complement of Russell's class). – Zvonimir Sikic Oct 6 at 9:38
  • @ZvonimirSikic: The "decent class" which you have described is ill-defined under ZFC (sets can't contain themselves, classes may only contain sets, therefore you can't have a self-referential structure at all) and NF (classes may still only contain sets). If instead you ask about the class of all sets which contain themselves, that's a perfectly reasonable object in NF, but it's not a set and (therefore) does not contain itself, because its formula is not stratified. – Kevin Oct 6 at 17:15
  • Kevin: This is true. But, just to mention, the point of my Zeitschrift article (cf. above), was to connect paradoxical classes with Cantor's theorem and to further relax restrictions on comprehension from stratification to "not in opposition to Cantor's theorem" and the decent class is paradoxical even in this relaxed sense. – Zvonimir Sikic Oct 7 at 9:41
  • @ZvonimirSikic: I have never heard of a version of set theory in which classes are allowed to contain (proper) classes. If that is really what you want to do, I think you need to explicitly axiomatize it, because otherwise I just don't think I can rigorously reason about the object you are describing. – Kevin Oct 7 at 20:03
  • In NF the universal class V contains itself. But there is a bigger problem with axiomatization of my "not in opposition to Cantor's theorem" as a more relaxed comprehension principle then stratification (which was pointed to me by Quine). It is not effectively testable and hence not directly amenable to rules of (first order) proofs. – Zvonimir Sikic Oct 8 at 23:04

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