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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)?

bautzeman
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  • Trivial logics are just that trivial. – Mozibur Ullah Sep 04 '18 at 00:47
  • In naive set theory, (using) classical logic, we **can** describe Russell's set and show that it produces a contradiction. – Mauro ALLEGRANZA Sep 04 '18 at 06:56
  • "is it there any logic system or any method that can provide this (to the Paradox) solution? " Yes: see [Axiomatic Set Theory](https://plato.stanford.edu/entries/set-theory/#AxiSetThe). – Mauro ALLEGRANZA Sep 04 '18 at 08:58
  • @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 04 '18 at 11:15
  • Already asked (and answered) twice... See also [Paraconsistent set theory](https://plato.stanford.edu/entries/logic-paraconsistent/#SetTheo) : "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 04 '18 at 11:17
  • @MauroALLEGRANZA and is "axiomatic set theory" "inside" or part of classical logic? but if in classical logic we cannot find/describe the solution to Russell's set paradox in naive-set-theory how could this be possible? And if it is not part of classical logic, then, could we find here the impossible solution that cannot exist in classical-logic-naive-set-theory for Russell's paradox? – bautzeman Sep 04 '18 at 11:20
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/82717/discussion-between-bautzeman-and-mauro-allegranza). – bautzeman Sep 04 '18 at 11:20
  • @MauroALLEGRANZA but a philosopher once told me that ZFC was not exactly the impossible solution that we cannot find/describe in classical logic and naive set theory. – bautzeman Sep 04 '18 at 15:05
  • @MauroALLEGRANZA "Given Zermelo-Fraenkel set theory (ZFC), Russell's set does not exist, and any object satisfying the description of Russel's Set is actually a set. Therefore, all statements regarding Russell's Set, given ZFC, are vacuously true at best, and imply nothing. This avoids the existence of Russell's Set from breaking logic itself. Scott-Potter set theory solves the paradox in the same way." – bautzeman Sep 04 '18 at 15:06
  • @MauroALLEGRANZA "I meant to say, Given ZFC.... no object satisfying the description of Russell's set is actually a set." And also, besides axiomatic and paraconsistent set theory, is it there any trivialism set theory? – bautzeman Sep 04 '18 at 17:42
  • Possible duplicate of [Is there any logical system/method where impossible/illogical/inconsistent things can exist (like a solution to Russell's paradox that makes sense)?](https://philosophy.stackexchange.com/questions/55023/is-there-any-logical-system-method-where-impossible-illogical-inconsistent-thing) This is a third post asking the same question, there is a reason we have rules against duplicates. – Conifold Sep 04 '18 at 19:55
  • @MoziburUllah And could these trivial logics provide this solution? I mean could we find here the impossible solution that cannot exist in classical-logic-naive-set-theory for Russell's paradox? – bautzeman Sep 04 '18 at 20:09
  • Nowadays, ordinary set theory and logic provide a solution to RP. It hasn't been an issue in mathematics for a century or more. A few zealots still seem to be hammering away at the so-called Liar Paradox, but their solution seems to be to accept certain inconsistencies in their system of logic. Sounds like a dead end to me. Or maybe it just needs a bit tweaking. You might look them up anyway, but LP never was an issue in mathematics AFAIK. – Dan Christensen Sep 11 '18 at 02:52

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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.

yuggib
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