  The universe of sets V is the union of all such stages: V=∪α∈On Vα, where On is the class of ordinals.  The first infinite level Vω consists of all of the hereditarily finite sets[3] and this level satisfies ZFC-Infinity. The sets at this level can be coded by natural numbers and in this way one can show that PA and ZFC-Infinity are mutually interpretable.[4] The second infinite level Vω+1 is essentially P(ℕ) (or, equivalently, ℝ) and this level satisfies (a theory that is mutually interpretable with) PA2.  The third infinite level Vω+2 is essentially P(P(ℕ)) (or, equivalently, as the set of functions of real numbers) and this level satisfies (a theory that is mutually interpretable with) PA3.  The first three infinite levels thus encompass arithmetic, analysis and functional analysis and therewith most of standard mathematics.  In this fashion, the hierarchy of sets and associated set-theoretic systems encompasses the objects and systems of standard mathematics.