> a proof that Er[ xer <--> ~xex ]. IF THAT HAPPENS, THEN YOUR THEORY > IS INCONSISTENT and SOME part of it MUST be abandoned.
So the AXIOMS are a *CONSTRAINT* what formula are allowed.
 > (P->Q)->((P^R)->Q) IS A TAUTOLOGY -- and there are MANY SIMPLE > "algorithms" that PROVE THAT THAT IS A TAUTOLOGY!
only in FOL.
Axioms (R here) are 2OL or 3OL (Regularity)
so they range over all expressions within their own theory (including P and Q) and LIMIT the usage/range of certain symbols.
 >You need ZFC to tell you what sets DO exist! -- This validity ONLY >tells you that a certain kind of set canNOT exist!
ALL(set) mem e set IFF p(mem,set) AND EXIST(anotherset) mem e anotherset
This is what ZFC does, limits the relation 'e'.
 > The way this paradox can thwart your set theory > IS IF your set theory, in proving that many sets exist, > HAPPENS TO STUMBLE ACROSS > a proof that Er[ xer <--> ~xex ]. > IF THAT HAPPENS, THEN YOUR THEORY IS > INCONSISTENT and SOME part of it MUST be abandoned. > In naive set theory, THIS DOES happen > (this set r MUST exist), so naive set > theory is inconsistent (and therefore worthless).
What's the difference between Predicate Calculus and Naive Set Theory.
They are both ZERO-AXIOM logic systems.
If NST is inconsistent, what rules in Predicate Calculus are you using to ensure ZFC does not inherit NST inconsistency?