1. Identical sets are identical.
2. Different sets are different.
3. Statements contradicting axioms 1 or 2 are false or malformed.
4. A malformed statement is one for which a conclusion does not follow given its structure.
5. A false statement is one that while structurally correct is not true.
The "structure" I mean has to do with how the sentence is put together.
For instance, the following is a badly structured syllogism:
If x=1, and y=2, then x=y.
The basic structure for the syllogism--the well-formed structure--is,
if a=b, and b=c, then a=c,
and variations on that structure are to be considered malformed.
Notice that with the first given that x does not equal y, the conclusion is false, but the entire statement is malformed so that is secondary.
With the basic axioms established--and notice how simple they are--it's trivial to handle supposed paradoxes which reduce to attacking one of the first two axioms.
For instance, the so-called Russell Paradox reduces to the assertion that a set includes and excludes itself.
Let A be a set that includes itself, and let B be a set that excludes itself.
B is different from A.
Therefore, by axiom 2 any statement that B is A is malformed or false.
Stating that B is A and B is different from A is structurally wrong, so the full statement is malformed.
Notice also that the resolution to the supposed paradox is the well-formed statement:
Consider a set A that includes all and only sets, except itself, that exclude themselves.
Notice that the axioms prevent that set from including itself, as then you reduce to a set that both includes and excludes itself, against axiom 2 as I explained.