Russell’s Paradox

Sets are collections of things. Under Unrestricted Comprehension, a set can be made with any property at all. For example, with the property of being a cat, you can have the set of cats: the set that has as its members anything that has the property of being a cat.

You can even evaluate the set itself with the property it is the set of. The set of cats is not a cat. And so, the set of cats is not a member of itself. What about the set of non-cats, of those things that are not a cat? The set of non-cats, the collection of all those things that aren’t cats, is itself not a cat. And so, the set of non-cats has itself as a member.

Some sets (the set of cats) do not include itself as a member while others (the set of non-cats) do. Russell’s paradoxical set is:

Let R = { x | x ∉ x }

This says: let set R be the set of anything x that is not a member of itself.

The set of cats is a member of R, since the set of cats has the property of non-membership of itself that is specified by R, since it’s not a member of itself. On the other hand, the set of non-cats is not a member of R, since the set of non-cats includes itself as a member, itself being not a cat, and thus does not have the property specified by R. Russell’s paradoxical query goes: what about R, does R have itself as a member or not?

Consider that R has itself as a member. However, R specifies sets that do not have themselves as a member. And so, R cannot be a member of itself. But maybe that’s the answer, and we made a discovery: R is not a member of itself! Unfortunately no. In this case, R must be a member of itself, since this is exactly what R specifies. The result is that if set R is a member of itself, then set R is not a member of itself, and if set R is not a member of itself, then set R is a member of itself; i.e. RRRR. This result of Russell’s Paradox can be written:

Let R = { x | xx }, then RRRR.

Combine this result with the truth of the Principle of Bivalence of Sets that R either is or is not a member of itself, then the contradiction that R both is and is not a member of itself is readily derivable. (RR & RR) follows from the mere construction of set R and the truth that RR v RR.

Historically, set theory has been amended in various ways, typically through axioms, so that R cannot be stated. I take it that all such solutions are similar for doing away with Unrestricted Comprehension, restricting how sets can be built from properties.

 

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