When Zermelo’s axioms 1–8 were found to be inadequate for a full-blown development of transfinite induction and ordinal arithmetic, Fraenkel and Skolem independently proposed an additional axiom schema to eliminate the difficulty. As modified by Hungarian-born American mathematician John von Neumann, it says, intuitively, that if with each element of a set there is associated exactly one set, then the collection of the associated sets is itself a set; i.e., it offers a way to “collect” existing sets to form sets. As an illustration, each of ω, P(ω), P(P(ω)), …, formed by recursively taking power sets (set formed of all the subsets of the preceding set), is a set in the theory based on Zermelo’s original eight axioms. But there appears to be no way to establish the existence of the set having all these sets as its members. However, an instance of the “axiom schema of replacement” (axiom 9 in the table) provides for its existence.

Intuitively, the axiom schema of replacement is the assertion that, if the domain of a function is a set, then so is its range. That this is a powerful schema (in respect to the further inferences that it yields) is suggested by the fact that the axiom schema of separation can be derived from it and that, when applied in conjunction with the axiom of power set, the axiom of pairing can be deduced.

The axiom schema of replacement has played a significant role in developing a theory of ordinal numbers. In contrast to cardinal numbers, which serve to designate the size of a set, ordinal numbers are used to determine positions within a prescribed well-ordered sequence. Under an approach conceived by von Neumann, if A is a set, the successor A′ of A is the set obtained by adjoining A to the elements of A (A′ = A ∪ {A}). In terms of this notion the natural numbers, as defined above, are simply the succession 0, 0′, 0″, 0‴, … ; i.e., the natural numbers are the sets obtained starting with Ø and iterating the prime operation a finite number of times. The natural numbers are well-ordered by the ∊ relation, and with this ordering they constitute the finite ordinal numbers. The axiom of infinity secures the existence of the set of natural numbers, and the set ω is the first infinite ordinal. Greater ordinal numbers are obtained by iterating the prime operation beginning with ω. An instance of the axiom schema of replacement asserts that ω, ω′, ω″, … form a set. The union of this set and ω is the still greater ordinal that is denoted by ω2 (employing notation from ordinal arithmetic). A repetition of this process beginning with ω2 yields the ordinals (ω2)′, (ω2)″, … ; next after all of those of this form is ω3. In this way the sequence of ordinals ω, ω2, ω3, … is generated. An application of the axiom schema of replacement then yields the ordinal that follows all of these in the same sense in which ω follows the finite ordinals; in the notation from ordinal arithmetic, it is ω². At this point the iteration process can be repeated. In summary, the axiom schema of replacement together with the other axioms make possible the extension of the counting process as far beyond the natural numbers as one chooses.

In the ZFC system, cardinal numbers are defined as certain ordinals. From the well-ordering theorem (a consequence of the axiom of choice), it follows that every set A is equivalent to some ordinal number. Also, the totality of ordinals equivalent to A can be shown to form a set. Then a natural choice for the cardinal number of A is the least ordinal to which A is equivalent. This is the motivation for defining a cardinal number as an ordinal that is not equivalent to any smaller ordinal. The arithmetics of both cardinal and ordinal numbers have been fully developed. That of finite cardinals and ordinals coincides with the arithmetic of the natural numbers. For infinite cardinals, the arithmetic is uninteresting since, as a consequence of the axiom of choice, both the sum and product of two such cardinals are equal to the maximum of the two. In contrast, the arithmetic of infinite ordinals is interesting and presents a wide assortment of oddities.

In addition to the guidelines already mentioned for the choice of axioms of ZFC, another guideline is taken into account by some set theorists. For the purposes of foundational studies of mathematics, it is assumed that mathematics is consistent; otherwise, any foundation would fail. It may thus be reasoned that, if a precise account of the intuitive usages of sets by mathematicians is given, an adequate and correct foundation will result. Traditionally, mathematicians deal with the integers, with real numbers, and with functions. Thus, an intuitive hierarchy of sets in which these entities appear should be a model of ZFC. It is possible to construct such a hierarchy explicitly from the empty set by iterating the operations of forming power sets and unions in the following way.

The bottom of the hierarchy is composed of the sets A₀ = Ø, A₁, …, A_n, …, in which each A_{n + 1} is the power set of the preceding A_n. Then one can form the union A_ω of all sets constructed thus far. This can be followed by iterating the power set operation as before: A_ω′ is the power set of A_ω and so forth. This construction can be extended to arbitrarily high transfinite levels. There is no highest level of the hierarchy; at each level, the union of what has been constructed thus far can be taken and the power set operation applied to the elements. In general, for each ordinal number α one obtains a set A_α, each member of which is a subset of some A_β that is lower in the hierarchy. The hierarchy obtained in this way is called the iterative hierarchy. The domain of the intuitive model of ZFC is conceived as the union of all sets in the iterative hierarchy. In other words, a set is in the model if it is an element of some set A_α of the iterative hierarchy.

Axiom for eliminating infinite descending species

From the assumptions that this system of set theory is sufficiently comprehensive for mathematics and that it is the model to be “captured” by the axioms of ZFC, it may be argued that models of axioms 1 through 9 of the table that differ sharply from this system should be ruled out. The discovery of such a model led to the formulation by von Neumann of axiom 10, the axiom of restriction, or foundation axiom.

This axiom eliminates from the models of the first nine axioms those in which there exist infinite descending ∊-chains (i.e., sequences x₁, x₂, x₃, … such that x₂ ∊ x₁, x₃ ∊ x₂, …), a phenomenon that does not appear in the model based on an iterative hierarchy described above. (The existence of models having such chains was discovered by the Russian mathematician Dimitry Mirimanoff in 1917.) It also has other attractive consequences; e.g., a simpler definition of the notion of ordinal number is possible. Yet there is no unanimity among mathematicians whether there are sufficient grounds for adopting it as an additional axiom. On the one hand, the axiom is equivalent (in a theory that allows only sets) to the statement that every set appears in the iterative hierarchy informally described above—there are no other sets. So it formulates the view that this is what the universe of all sets is really like. On the other hand, there is no compelling need to rule out sets that might lie outside the hierarchy—the axiom has not been shown to have any mathematical applications.