Publisher Summary This chapter describes the simple theory of types and some of its extensions. The simple theory of types is certainly one of the most natural systems of set theory; the reason for this is that it is defined rather by a family of structures than by a system of axioms. The axioms of extensionality and of comprehension are in a certain sense the same for all types. The chapter consistently adds to the ideal type theory. “Ideal type theory” could now be defined as the set of sentences (formulas without free variables) holding in every structure (T0, T1,…; =, Є), where Tk+l is the power set of Tk and where “=,” “ Є,” and the variables are interpreted in the obvious way. One group of axioms of type theory is the axioms of extensionality. There is one such axiom for every type (except type 0), and it says that two sets are equal if they have the same elements.