Set Theory and Beyond

 I promised to talk about boolean algebras and I'll do that now. Instead of going for the full generality, let's stick with the picture drawn last time. We'll fix a set $U$ which is not empty and look at its subsets. Just for today, we'll call these subsets objects . So saying that $X$ is an object just means that $X\subseteq U$. There are three binary operations between objects that we are interested in. "Binary" here means that the operation takes two objects as its arguments and returns a third one as the result. As you guessed it, those binary operations are $\cup ,\cap ,-$. We'll also consider a unary operation ${}^{\complement }$, i.e. one that takes a single argument $X$ and returns the $X^{\complement }:=U-X$. Note here that this operation ("the complement") only makes sense if we fix an "all-enveloping set" $U$. Otherwise, the complement would be something like $$\{x:x\notin X\}$$ which is not a set! In addition to ...