- Axiom 3.8. (Univeral specification). (Dangerous!) Suppose for every object \(x\) we have a property \(P(x)\) pertaining to \(x\) (so that every \(x, P(x)\) is either a true statment or a false statement). Then there exists a set \(\{x:P(x)\text{ is true}\}\) such that for every object \(y\), $$ y\in\{x:P(x)\text{ is true}\}\iff P(y)\text{ is true}. $$
This axiom is also known as the axiom of comprehension. It asserts that every property corresponds to a set; if we assumed that axiom, we could talk about the set of all blue objects, the set of all natural numbers, the set of all sets, and so forth.
Russell's paradox $$
\Omega:=\{x:x\text{ is a set and }x\notin x\}
$$ $$
\Omega\in\Omega?
$$