Here's the flaw in your argument...
It comes at the point where you consider an arbitrary false statement and try to show that it is a member of the set of true statements. (Nothing wrong with doing that in logic, if you can pull it off, but here you just can't...)
If a statement is false, then it implies a contradiction (as we find in a proof by contradiction). Since anything follows from a contradiction, it follows that the statement is true. Thus the statement is a member of the set of true statements.
Okay you're considering the case of a statement that is false.
In "proof by contradiction", an assumption that a false statement is true
leads to a contradiction, from which in turn follows any conclusion you like, including your own desired conclusion in this thread that our false statement is a member of the set of true statements.
But we are not
assuming that our false statement is true; we are considering the case where it is false. A contradiction does not
follow from an assumption we haven't made about our false statement being true. We may not
conclude anything we like from a contradiction that we haven't derived. Specifically, we cannot conclude that the false statement under consideration is a member of the set of true statements.
Sorry, friendo, but your argument doesn't demonstrate that there is any kind of paradox associated either with proof by contradiction or with axiomatic set theory.
It does, however, demonstrate some clever sleight of hand.