TITLE: What are the consequence of negating all ZF axioms?
QUESTION [5 upvotes]: This is not a question about "if a false statement with ZF(C) is considered true, what happens." (The answer for that is trivial, as falsity in classical logic proves everything.) What I am rather asking is what happens if we negate axioms of ZF(C) - in other words, axioms of the new set theory, call it $ZF(C)'''$, are exact negation of each axiom of ZF(C).
When I think of such $ZF'''$ (I am going to drop axiom of choice being negated for now), it has to mirror $ZF$ in some way, so I would expect that all false statements with ZF would now be true statements with $ZF'''$ and true statement with ZF would now seem to be false statement in $ZF'''$. Yet this does not seem to be true also, because I really have not negated laws of natural deduction system in propositional logic, contained in first-order logic.
What really then would be consequences of $ZF'''$?

REPLY [4 votes]: Well, the big change (as Clive Newstead points out in the comments above) is that it stops being what one could intuitively call a "set theory." One could certainly call it a "model theory," but it stops acting anything like what we'd expect of sets. Let's just deal with the immediate consequences.

Since Extensionality is negated, then there exist "sets" $x$ and $y$ with exactly the same "elements," but such that $x\neq y.$ What are "sets" and "elements" that this should be so? Or, perhaps, what does "=" mean that this should be so?
Since Replacement is negated, then there is a "set" $x$ and a "function" $f$ on $x$ such that the "image" of $f$ is not a "set," but a "proper class." But typically, a proper class is a class that is (in a sense) too big to be a set, but images of functions are intuitively no larger than the domains, so images of functions on sets won't be too big to be sets. So what in the world are "sets"/"functions"/"images"/"proper classes"?
Since Comprehension is negated, then there is a "set" $x$ and a formula $\phi$ such that the "class" of "elements" of $x$ satisfying $\phi$ is not a "set," but a "proper class." Why?
Since Pairing is negated, then there exist "sets" $x$ and $y$ such that any "class" having both $x$ and $y$ as "elements" is a "proper class." Why?
Since Power Set is negated, then there is a "set" $x$ such that any "class" containing all "subsets" of $x$ is a "proper class." Why?
Since Union is negated, then there is a "set" $x$ whose "elements" are all "sets," but such that any "class" containing all "elements" of the "elements" of $x$ is a "proper class." Why?
Since Infinity is negated, then either $\emptyset$ isn't a "set," or
$\emptyset$ is a "set" but the von Neumann ordinal $\omega$ is not a "set." The latter isn't a huge deal, but why should the former be true?
Since Foundation is negated, then there is a "non-empty" "set" $x$ such that every "element" of $x$ has an "element" in common with $x.$ Again, this isn't too objectionable.