Russell's Paradox and Open Question Arguments

This is probably very obvious to people who think about this more than I do.

Gödel's proofs show that any attempt to reduce mathematics to logic would generate a kind of open question argument, because any method of demonstrating the validity of a theory in the relevant logic would have an arithmetical isomorph.

Russell's paradox shows the impossibility of reducing mathematics to the most unrestricted kind of set theory (Frege's project), because the naïve concept of 'set' which it employs is inconsistent.  There are sets whose intensional definitions generate ambiguous extensions.

There are similarities between the open question paradoxes and the class paradoxes, and the attempts to address them have a lot in common with one another.  Stipulation, hierarchy, and concrete construction rules figure in both.

Russell's paradox is generated because we assume that our concept of a set is consistent.  But we can only prove this by having a method of distinguishing sets - and this method, if it had an arithmetical isomorph, would depend upon the undefined notion of a set on which some logics of arithmetic depend.  Also, if we could construct a logic of arithmetic without the concept of a set, then Russell's paradox - and indeed the whole of set theory - would no longer be important.  Gödel has shown, however, that no other approach to logification can avoid replacing sets with some other paradoxical fundamental.

The 'open question' in set theory is:  'What is a set?'.   In other words:  'What kinds of things are included in the set of sets?'

We can't answer this question without circularity (and so Russell's paradox) or stipulation (various type theories, ZF set theory etc.).  We can answer it recursively if we can show that some (necessarily 'open') concept of a set or class is grammatically necessary (in the Wittgensteinian sense - i.e. necessary if we are to talk at all).

Stipulative definitions are always incomplete, because we can't render the language of the stipulation entirely unambiguous.  A stipulative definition which it would be unintelligble to query is a recursive definition.