Set Theory and Beyond

We have already talked about properties, but we didn't define them. I'm not going to that right now either, but I remind you that a property (or a predicate) is some definite way to separate sets to those which have that property and to those which do not. For example, if $P(x)$ is a property and $a$ is a set, then $a$ might have the property $P$, written as $P(a)$, or it might not have the property $P$, written as $\mathrm{\neg }P(a)$. Now, for any set $A$, we may look at the property $S_A(x)$ asserting "$x$ belongs to $A$". In this way, every set can be seen as a property, but the other way around is not true. An example of the property which doesn't come from a set is given by so-called Russell's paradox.  Namely, consider the property $R(x)$ asserting that "$x$ does not belong to $x$". I claim that this property does not come from a set, or more precisely, that there is no set $r$ such that the property $R$ is the same as t...