Did Russell really find a contradiction in Set Theory?

[molbol2000]

The Russell set formula is inconsistent. But almost every language allows for contradictory or incorrect but grammatically correct formulas. For example, the arithmetic expression 1 + 1 = 5 is incorrect and inconsistent. Thus, Russell proved not the inconsistency of set theory (Cantor's), but only that the language allows for incorrect expressions

[HallsofIvy]

You appear to be misunderstanding "inconsistency".  It does not mean that you can state an incorrect formula.    It means that you can prove, from the basic axioms and definitions,  two contradictory statements.  In "naive" set theory, a basic axiom is that "if you can define a set rigorously, it exists".

Therefore, "the set of all sets that do not contain themselves", since we can, naively, look at the definition of any set and determine whether or not it contains itself, must exist.

But once we have decided that it is a set we can ask whether or not it contains itself.   It cannot contain itself because that contradicts the fact that it only contains sets that do NOT contain themselves.   But if it does not contain  itself then it cannot be said that it contains allsets that do not contain themselves.