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Consider the following definition of set given by Cantor in a letter to Dedekind:

If on the other hand the totality of the elements of a multiplicity can be thought of without contradiction as 'being together', so that they can be gathered together into 'one thing' I call it a consistent multiplicity or a 'set'.
("Letter to Dedekind" (1899), in From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, edited by Jean van Heijenoort, pg 114)

If one precisely formulated this principle in the formal language of set theory and added it as a axiom (or even used as-is as a 'rule-of-thumb' of set formation) to (for) the following version of Naive Set Theory:

Extensionality: Given two sets A and B , A=B iff (x)[x 'is a member of' A iff x is a member of' B]

Comprehension: Given any predicate P(x), the set {x|P(x)} exists and (a)[a 'is a member of' {x| P(x)} iff P(a)]

Can one still derive paradoxes from this amended version of Naive Set Theory? (In using this definition as a 'rule-of-thumb' for set formation one seeks to gather together the elements of the multiplicity into 'one thing' in such manner as to avoid the apparent paradox.)

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The axiom of comprehension allows one to create a contradiction by the Russell's paradox. (A contradiction, supposing you have some axioms implying/stating that any member of the universe either is or is not a member of a given set.)

The quote presents another means of creating sets - namely, taking elements so that grouping them can't lead to a contradiction allows one to create a set.

Another interpretation of the quote would be to take it as a definition of set. The definition is different from definition of set in the naive set theory, which does lead to contradiction. In other words, the sets in naive set theory are not sets in the sense the quote defines.

Furthermore, the quote provides a very poor definition or axiom, since verifying if something is a set would require checking if it leads to contradiction. You would have to define contradiction in some way that one can check within the axiomatic system one is using. How would one do this so that one can still prove that e.g. finite unions of sets are still sets, or that power sets of sets are sets?

  • Let's consider the multiplicity from which Russell's Paradox is defined (initially the complex symbol {x|P(x)} should be deemed a multiplicity): V={x| x is a set and x 'is not a member of' x}. Now – Thomas Benjamin May 10 '12 at 5:07
  • One notes that V does form a set given that '___ is a member of V' and not-'____ is a member of V' cannot be predicated of V (after all, the universe V of ZF set theory contains foundational sets, i.e. sets that are not members of themselves) so Comprehension and Cantor's 'rule-of-thumb are both satisfied. Similarly for Curry's paradox (the set {x|x 'is a member of' x ----->F and F is false} does not exist) and the Burali-Forti and Cantor's Paradox ( the sets ORD and U={x| x is a set and x=x} both do not satisfy the Axiom of Foundation and are non-wellfounded sets since Ord and U – Thomas Benjamin May 10 '12 at 5:31
  • can be members of themselves and this dissolves the Burali-Forti and Cantor's paradoxes). My first comment should have gone as follows: Now the question is asked, is V a member of V? Obviously, '___ is a member of V' and not-'___ is a member of V' cannot be consistently predicated of V and that dissolves Russell's Paradox. – Thomas Benjamin May 10 '12 at 5:39
  • It should be noted that if one considers the theory of sets as a Science which studies certain entities i.e. sets then Cantor's 'rule-of-thumb' for set creation certainly augments Comprehension and such moves as I have made in dissolving Russell's Paradox and suggested in dissolving Curry's, Burali-Forti's, and Cantor's paradoxes are certainly valid. This, in my opinion, is the correct way to proceed in Naive Set Theory. – Thomas Benjamin May 10 '12 at 6:04
  • In paraconsistent set theory the Russell Set & Universal set can be defined. And it follows as a theorem that the union of all elements of the Russell Set is the Universal Set. – Mozibur Ullah Feb 6 '14 at 9:11
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Of course there are paradoxes persisting. But always when in history a paradox or contradiction has raised its ugly head its derivation has been denounced as invalid by orthodox set theorists. The Löwenheim-Skolem paradox has been explained away by different notions of countability in "inner" and "outer" models. The antinomies raised by Vitali, Hausdorff, and Banach-Tarski persisting in axiomatic set theory have been "resolved" by declaring sets used for the contradictions as not measurable. And so on.

A multitude of examples is analyzed in Mueckenheim's comprehensive Source Book on Transfinity https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf. There are also more than 200 opinions collected which usually are silenced by massive downvoting and deleting in forums dominated by set theorists like MathOverflow or MathStackExchange.

  • Are you just pointing out the problems with doing set theory "naively"? The way you've phrased things make it sound like you think ZFC is inconsistent, along with various other disciplines (e.g. model theory and measure theory). – user6559 Jul 22 '17 at 6:56
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    @Hurkly: ZFC is inconsistent, at least with mathematics. See the formal proof in hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf on p. 258: Not enumerating all positive rational numbers (formal proof). Mathematics gives the limit inifinity for the not enumerated fractions, ZFC gives the limit empty set. Or take, as an easy example, McDuck on p. 235 loc cit. According to mathematics he becomes infinitely rich, ZFC let him go bankrupt. – Heinrich Jul 22 '17 at 10:50
  • Ah, so it's less about being "naive", and less about "set theory" and more about simply getting wrong the process of formalizing one's ideas. – user6559 Jul 22 '17 at 15:43
  • @Hurkyl: Formalizing nonsense or formalizing in a nonsensical manner does not help. Formalizing McDuck's wealth and ending up with his bankruptcy shows a lack of thinking capability. You "formalize" that every dollar will be spent, but you forget that every dollar is followed by infinitely many. Fallacy of one-sided perspective. – Heinrich Jul 22 '17 at 21:21
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The amended version of set theory yields a lot of paradoxes. For an overview see chapter III of https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf. To expose only one of the most striking paradoxes (loc cit p. 253):

Every day Scrooge McDuck earns 10 enumerated dollars and spends 1 enumerated dollar. If he always returns the smallest number then the set theoretic limit of the sequence of his possessions is the empty set because from every dollar we can say when it is spent. Although his wealth grows beyond all limits, after all he will become bankrupt.

Certainly this will be considered as a paradox by every normal observer, but it is the foundation of naive as well as of modern set theory. It is required to defend the claim that all rational numbers can be enumerated.

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