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In naive set theory in classical logic, we cannot describe or find a solution to Russell's set paradox (it is impossible).

But is it there any logic system or any method that can provide this solution? Is it there any logic system or method where we could find and describe this solution? Would trivialism do the work (since contradictions and impossible things are allowed there)?

  • Trivial logics are just that trivial. – Mozibur Ullah Sep 4 '18 at 0:47
  • In naive set theory, (using) classical logic, we can describe Russell's set and show that it produces a contradiction. – Mauro ALLEGRANZA Sep 4 '18 at 6:56
  • "is it there any logic system or any method that can provide this (to the Paradox) solution? " Yes: see Axiomatic Set Theory. – Mauro ALLEGRANZA Sep 4 '18 at 8:58
  • @MauroALLEGRANZA But in a logic system where impossible things can happen like in trivialism couldn't we find the impossible solution that cannot exist in classical-logic-naive-set-theory for Russell's paradox (since it is impossible to find, wouldn't we be able to find it in a logic system where impossible things are valid or can be found, like in trivialism)? – bautzeman Sep 4 '18 at 11:15
  • Already asked (and answered) twice... See also Paraconsistent set theory : "The naive, and intuitively correct, axioms of set theory are the Comprehension Schema and Extensionality Principle. [...] It then follows that r∈r ∧ r∉r. A paraconsistent approach makes it possible to have theories of sethood in which the mathematically fundamental intuitions about these notions are respected. There are several approaches to set theory with naive comprehension via paraconsistent logic." – Mauro ALLEGRANZA Sep 4 '18 at 11:17
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The idea is to consider the collection of all sets as another type of object.

Usually, such objects are called classes. Bernays-Gödel set theory is a (conservative extension of ZFC) theory that includes classes, and where therefore the class of all sets is a well-defined concept.

Clearly, the class of all classes would have the same problems as the set of all sets, but this is avoided by the fact that in BG it is not possible to quantify over classes.

Professional mathematicians that do not work in logic or set theory, i.e. most of them, take a more relaxed approach to classes, and mostly use them as semi-rigorous objects, being careful not to quantify over them but using them essentially as sets. One such example is category theory, where many of the categories commonly used are classes.

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