Let F(x) be a formula belonging to the language of first order ZF in which x is the one and only
variable that occurs free and let N(x) be the negation of F(x). Are any examples known of an F(x)
and an N(x) such that not just one but each one of the two sentences "The set (x:F(X)) exists" and
"The set (x:N(x)) exists"-which can be expressed in the language of first order ZF-are inconsistent
with ZF? An obvious candidate for F(x) is the formula which defines the Russell set but in that case
N(x) defines the null set (because of the Axiom of Foundation in ZF) and there is no inconsistency.



A question about "paradoxical" sentences in the language of ZF set theory.
Here are two examples:


1. Let $F(x)$ be the statement "There exists $y$ such that $x$ is the power set of $y$", or formally, $$F(x)=\exists y\forall u(u\subseteq y\leftrightarrow u\in x).$$


Since every set has a (unique) power set, we have that the class of power sets is not a set; but also the class of those which are not power sets is not a set either, because if $x=\mathcal P(y)$ then $x\setminus\lbrace\varnothing\rbrace$ is not a power set.
2. Let $F(x)$ be the statement "$x$ is an ordinal". It is known that the class of ordinals is not a set, but the class of those which are not ordinals is not a set either, e.g. $\mathcal P(\alpha)$ for every ordinal $\alpha$, is not an ordinal (with the exceptions of $0$ and $1$ of course).


