> I can if there's no rule of inference.  Perhaps that's crux.  You are 
> requiring
> that a "mathematical structure" be a set of axioms *plus* the usual rules of
> inference for "and", "or", "every", "any",...and maybe the axiom of choice 
> too.

Rules of inference can be derived from the axioms...it sounds circular
but in ZFC all you need are nine axioms and two undefinables (which
are set, and the binary relation of membership).  You write the axioms
using the language of predicate calculus, but that's just a
convenience to be able to refer to them.

> Well not  entirely by itself - one still needs the rules of inference to get 
> to
> Russell's paradox.

Not true!  The paradox arises from the axioms alone (and it isn't a
true paradox, either, in that it doesn't cause a contradiction among
the axioms...it simply reveals that certain sets do not exist).  The
set of all sets cannot exist because it would contradict the Axiom of
Extensionality, which says that each set is determined by its elements
(something can't both be in a set and not in the same set, in other


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