Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule (Part. I)

Abstract In this article Russell’s paradox and Cantor’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.

https://www.journaljamcs.com/index.php/JAMCS/article/view/30339

The paradox is a consequence of the unrestricted form of comprehension found in naive set theory. The solutions to the paradox involve the restriction of comprehension by various schemes including the use of type hierarchies, stratification and restricting comprehension to range over already defined sets.

The existence of the paradox however does rule out the existence of so called large sets or classes, such as the set of all sets, but does mean they must be handled with great care.

One view is that the von Neumann-Gödel-Bernays or Kelley-Morse systems are the "natural" correct versions of set theory and that Russell's paradox was an obvious flaw of previous systems which had an equally obvious correction (this was Gödel's view). This correction involves the distinction between a set and a class. The idea is that if a class A belongs to a class B then this is also saying something about the nature of A, that A must be a "smaller", more restricted kind of class, namely a set. The Russel class R of all classes which do not belong to themselves is then shown not to be set and thus not to be a member of itself.

The need to distinguish classes and sets turns up in category theory in the guise of "small" and "large" categories. However Grothendieck used another approach to solve this problem (Tarski-Grothendieck set theory) which involves fixed "universes" containing inaccessible cardinals.

As I recall, von Neumann addressed the Russell Paradox with sweeping and definitive changes to Set Theory.

I think that here there is a confusion of levels of reality. Things are not, per se, members of any set. The set is an intellectual artifact used for the classification of things. The question of whether a set is a member of itself or not is meaningless. As Marx (Groucho) said: "I'd never belong to a club that had me as one of its members"

The question about whether there is a paradox can be ambiguous if we do not take into account that we are talking about theories, about texts. As such, said "paradox" is a genuine contradiction -sometimes also called antinomy- that arises from the ontological problem of assuming -within a certain version of set theory-, that there are sets that contain themselves as elements and others not. Russell's solution is, in my opinion, an ad hoc "solution", because it avoids the contradiction at the price of exchanging the ontology for one that cannot be independently justified.

I think the problem with any set theory - a genuine philosophical and not a logical problem - is that some authors assume that "set" is an undefined term, but others identify it with "collection" (but this cannot explain the existence of empty sets without changing the meaning of the word "collection").

I understand that the problem behind the paradox is one already addressed by the nominalist tradition and by Wittgenstein in the Tractatus, respect to the presumed relation of identity of a thing with itself : according to the theory there are sets that include themselves as elements but, does it make sense to say that a thing is related to itself, or for there to be a relation there must be at least two relata or terms of the relation?

"Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all" (Wittgenstein, 1921, 5.5323)

Note that stupid aristotelian logic cannot ansfered this simple question

The barber is the "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself?

Contemporary set theoretists

https://scholar.google.com/citations?view_op=search_authors&hl=ru&mauthors=label:set_theory

says that such barber cannot exists, since there is known Russells contradiction.

Of course that is dumb ansver from a dumb persons.

Russell's barber says that: something wrong with classical logic