Russell's paradox and classes
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 \(\neg P(a)\). 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...