TITLE: Is the Russell paradox the only possible contradiction to the axiom schema of comprehension due to Frege (1893)? $\{x:P(x)\}$
QUESTION [5 upvotes]: Is the Russell paradox the only possible contradiction to the axiom schema of comprehension due to Frege (1893)? The axiom that says that
if $\varphi$ is a property, then there exists a set 
$Y = \{X: \varphi(X)\}$ of all
elements having property $\varphi$.
If not then what are other paradoxes that result from that axiom?

REPLY [5 votes]: In addition to the paradoxes mentioned by Asaf, there is also the infinite family:
$$\begin{align}
\varphi(x) & = \lnot\exists y.x\in y \land y\in x\\
\varphi(x) &= \lnot\exists y\exists z.x\in y \land y\in z \land z\in x\\
&\vdots\end{align}$$
Somewhat different is Curry's paradox: $$\varphi_Y(x) = (x\in x)\to Y$$Then following the same pattern of reasoning that gives Russell's paradox (that is, let $X = \{x\mid \varphi_Y(x)\}$ and ask if $X\in X$), in this case we can conclude $Y$.  But since $Y$ was completely arbitrary, we can prove anything at all by this method.