Beyond Meta-Predicate Logic: Set Theory

Set theory is a powerful tool in the logic of mathematics and other abstract areas of philosophical study. In set theory, there are two types of entities: elements and sets. In pure set theory, there are only sets; this is possible because sets can be elements of other sets and there is a such thing as the empty set. We will discuss pure set theory first. There is one relation in set theory or set theoretic operator which is that of membership. You can construct another set theoretic operator which expresses the concept of subsethood. A set, S, is a subset of another set, T, if and only if every member of S is a member of T (this also means that the empty set is a subset of every set). Two sets are identical just in case each is a subset of the other. Another set theoretic operator you can construct is called the powerset relation. A set, S, is the powerset of another set, T, if and only if S is the set of all subsets of T. Interestingly, the powerset of a set is always strictly larger than the set itself. For instance, the powerset of the empty set, {}, is the set which contains only the empty set {{}}; the powerset of the set which contains only the empty set is the set which contains only the empty set and the set which contains only the empty set {{},{{}}}. Etc. You can continue the powerset relation ad infinitum, constructing (or discovering) an infinite hierarchy of infinities.