Hi
I must admit I have not read much about this, but I can give my two cents. From what I have understand when you make a theory with axioms, you must allways be sure that the next axioms and definitions does not contradict the earlier ones. Also logic is just a tool for what your theory, I do not think you can say it is a part of the theory.
So set-theory is based on a few axioms, and the logic is the tool that are used to build set-theory. You can try to say that there can be a set that contains all other sets, because you allready have an axiom called the "axiom of specification", which allows you to make the subsets used in Russels paradox. But this implies the contradiction.
Also take this with a grain of salt. But I think that Russels paradox shows the contradiction with the axiom of specification. Because if you can make a subset of A where the elements of P(x) is true, where x are elements of A, you can also make a subset of B where ~P(x) is true. And using the rules of logic every elements in the main set must be in one of these subsets. But the set in Russels paradox is in none, hence it contradicts the axiom of specification.
from wikipedia:
"An axiomatic system is said to be
consistent if it lacks contradiction, i.e. the ability to derive both a statement and its negation from the system's axioms."
"An axiomatic system will be called
complete if for every statement, either itself or its negation is derivable."
http://en.wikipedia.org/wiki/Axiomatic_system#Properties
Maybe it is complete as you say, but not consistent.
