>Formal language solves paradoxes
I agree that formal languages often make apparent 'paradoxes' go away by being clear about what's meant, but if the paradox is an antinomy (like russell's paradox) then it just helps to make that clear and doesn't solve the question of "what should my language/axioms be?", since clearly they need to change to become consistent. Formal systems can of course provide a wealth of information about what exactly is wrong with one's assumptions, and that does help to fix them and make them consistent after an inconsistency is discovered.
Formal languages can have antinomys too, they aren't special in that regard.
>formal languages have indirectly taken over for math
Math can be viewed as a formal language, it's just that mathematicians barely ever bother to do it formally. Coq is a good example of doing it formally. I hope something like the https://en.wikipedia.org/wiki/QED_project
takes off once proof assistants become better.
It is work like solomonoff induction which takes significant ground from philosophy and brings rigor to it, and that wasn't done formally.
>For example, the "Barber Paradox" is a result of confusion of classes of different order, as is obvious to anybody who knows C++ or any other OO language.
I'd be interested to see this argument written out since I don't see a clear relationship between object hierarchies and sets. How are you converting sets to classes (or class hierarchies, etc) for the sake of making this argument?
How would you represent "the set which contains only itself" in this way? If it isn't possible under your system, then it is weaker than many set-theories and no replacement for those theories. (you may be interested in https://en.wikipedia.org/wiki/Homotopy_type_theory
since it takes this sort of apporach but I don't know anything about it)
OP didn't say it was.
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