Zermelo–Fraenkel Set Theory - Motivation Via The Cumulative Hierarchy

Motivation Via The Cumulative Hierarchy

One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von Neumann (Shoenfield 1977, sec. 2). In this viewpoint, the universe of set theory is built up in stages, with one stage for each ordinal number. At stage 0 there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2; see Hinman (2005, p. 467). The collection of all sets that are obtained in this way, over all the stages, is known as V. The sets in V can be arranged into a hierarchy by assigning to each set the first stage at which that set was added to V.

It is provable that a set is in V if and only if the set is pure and well-founded; and provable that V satisfies all the axioms of ZFC, if the class of ordinals has appropriate reflection properties. For example, suppose that a set x is added at stage α, which means that every element of x was added at a stage earlier than α. Then every subset of x is also added at stage α, because all elements of any subset of x were also added before stage α. This means that any subset of x which the axiom of separation can construct is added at stage α, and that the powerset of x will be added at the next stage after α. For a complete argument that V satisfies ZFC see Shoenfield (1977).

The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and related axiomatic set theories such as Von Neumann–Bernays–Gödel set theory (often called NBG) and Morse–Kelley set theory. The cumulative hierarchy is not compatible with other set theories such as New Foundations.

It is possible to change the definition of V so that at each stage, instead of adding all the subsets of the union of the previous stages, subsets are only added if they are definable in a certain sense. This results in a more "narrow" hierarchy which gives the constructible universe L, which also satisfies all the axioms of ZFC, including the axiom of choice. It is independent from the ZFC axioms whether V = L. Although the structure of L is more regular and well behaved than that of V, few mathematicians argue that V = L should be added to ZFC as an additional axiom.