Axiom schema of replacement

In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZFC) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZFC.

The axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is "small enough" to be a set, and there is a bijection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable bijections, which are identified with their defining formulas.

Contents

Statement

Suppose P is a definable binary relation (which may be a proper class) such that for every set x there is a unique set y such that P(x,y) holds. There is a corresponding definable function FP, where FP(X) = Y if and only if P(X,Y); F will also be a proper class if P is. Consider the (possibly proper) class B defined such for every set y, y is in B if and only if there is an x in A with FP(x) = y. B is called the image of A under FP, and denoted FP(A) or (using set-builder notation) {FP(x) : x ∈ A}.

The axiom schema of replacement states that if F is a definable class function, as above, and A is any set, then the image F(A) is also a set. This can be seen as a principle of smallness: the axiom states that if A is small enough to be a set, then F(A) is also small enough to be a set. It is implied by the stronger axiom of limitation of size.

Because it is impossible to quantify over definable functions in first-order logic, one instance of the schema is included for each formula φ in the language of set theory with free variables among w1, ..., wn, A, x, y; but B is not free in φ. In the formal language of set theory, the axiom schema is:

\begin{align} \forall w_1,\ldots,w_n \, \forall A \, ( [ \forall x \in A &\, \exists�! y \, \phi(x, y, w_1, \ldots, w_n, A) ] \\ &\Rightarrow \exist B \, \forall y \, [y \in B \Leftrightarrow \exist x \in A \, \phi(x, y, w_1, \ldots, w_n, A) ] ) \end{align}

Axiom schema of collection

The axiom schema of collection is closely related to and frequently confused with the axiom schema of replacement. While replacement says that the image itself is a set, collection merely says that a superclass of the image is a set. In other words, the resulting set, B, is not required to be minimal.

This version of collection also lacks the uniqueness requirement on φ. Suppose that the free variables of φ are among w1, ..., wn, x, y; but neither A nor B is free in φ. Then the axiom schema is:

\forall w_1,\ldots,w_n \,[(\forall x\, \exist\, y \phi(x, y, w_1, \ldots, w_n)) \Rightarrow \forall A\, \exist B\, \forall x \in A\, \exist y \in B\, \phi(x, y, w_1, \ldots, w_n)]

That is, the relation defined by φ is not required to be a function — some x in A may correspond to multiple y in B. In this case, the image set B whose existence is asserted must contain at least one such y for each x of the original set, with no guarantee that it will contain only one.

The axiom schema is sometimes stated without any restrictions on the predicate, φ:

\forall w_1,\ldots,w_n \, \forall A\, \exist B\,\forall x \in A\, [ \exists y \phi(x, y, w_1, \ldots, w_n) \Rightarrow \exist y \in B\,\phi(x, y, w_1, \ldots, w_n)]

In this case, there may be elements x in A that are not associated to any other sets by φ. However, the axiom schema as stated requires that, if an element x of A is associated with at least one set y, then the image set B will contain at least one such y. The resulting axiom schema is also called the axiom schema of boundedness.

The axiom schema of collection is equivalent to the axiom schema of replacement over the remainder of the ZF axioms.

Example applications

The ordinal number ω·2 = ω + ω (using the modern definition due to von Neumann) is the first ordinal that cannot be constructed without replacement. The axiom of infinity asserts the existence of the infinite sequence ω = {0, 1,2,...}, and only this sequence. One would like to define ω·2 to be the union of the sequence {ω, ω + 1, ω + 2,...}. However, arbitrary classes of ordinals need not be sets (the class of all ordinals is not a set, for example). Replacement allows one to replace each finite number n in ω with the corresponding ω + n, and guarantees that this class is a set. Note that one can easily construct a well-ordered set that is isomorphic to ω·2 without resorting to replacement – simply take the disjoint union of two copies of ω, with the second copy greater than the first – but that this is not an ordinal since it is not totally ordered by inclusion.

Clearly then, the existence of an assignment of an ordinal to every well-ordered set requires replacement as well. Similarly the von Neumann cardinal assignment which assigns a cardinal number to each set requires replacement, as well as axiom of choice.

Every countable limit ordinal requires replacement for its construction analogously to ω·2. Larger ordinals rely on replacement less directly. For example ω1, the first uncountable ordinal, can be constructed as follows – the set of countable well orders exists as a subset of P(N×N) by separation and powerset (a relation on A is a subset of A×A, and so an element of the power set P(A×A). A set of relations is thus a subset of P(A×A)). Replace each well-ordered set with its ordinal. This is the set of countable ordinals ω1, which can itself be shown to be uncountable. The construction uses replacement twice; once to ensure an ordinal assignment for each well ordered set and again to replace well ordered sets by their ordinals. This is a special case of the result of Hartogs number, and the general case can be proved similarly.

The axiom of choice without replacement (ZC set theory) is not strong enough to show that Borel sets are determined; for this, replacement is required.

History and philosophy

The axiom schema of replacement was not part of Ernst Zermelo's 1908 axiomatisation of set theory (Z); its introduction by Adolf Fraenkel in 1922 is what makes modern set theory Zermelo-Fraenkel set theory (ZF). The axiom was independently invented by Thoralf Skolem later in the same year. Although it is Skolem's final version of the axiom list that we use today, he usually gets no credit since each individual axiom was developed earlier by either Zermelo or Fraenkel.

The axiom schema of replacement drastically increases the strength of ZF, both in terms of the theorems it can prove and in terms of its proof-theoretic consistency strength, compared to Z. In particular, ZF proves the consistency of Z, as the set Vω·2 is a model of Z constructible in ZF. (Gödel's second incompleteness theorem shows that each of these theories contains a sentence, "expressing" the theory's own consistency, that is unprovable in that theory, if that theory is consistent (this result is often loosely expressed as the claim that neither of these theories can prove its own consistency, if it is consistent, but the work of Solomon Feferman [1960] shows that care must be taken in stating the content of the second incompleteness theorem.) The cardinal number \aleph_\omega is the first one which can be shown to exist in ZF but not in Z.

The axiom schema of replacement is not necessary for the proofs of most theorems of ordinary mathematics. Indeed, Zermelo set theory already can interpret second-order arithmetic and much of type theory in finite types, which in turn are sufficient to formalize the bulk of mathematics. A notable mathematical theorem that requires the axiom of replacement to be proved in ZF is the Borel determinacy theorem.

The axiom of replacement does have an important role in the study of set theory itself. For example, the replacement schema is needed to construct the von Neumann ordinals from ω·2 onwards; without replacement, it would be necessary to find some other representation for ordinal numbers.

Although the axiom schema of replacement is a standard axiom in set theory today, it is often omitted from systems of type theory and foundation systems in topos theory.

Relation to the axiom schema of specification

The axiom schema of specification, the other axiom schema in ZFC, is implied by the axiom schema of replacement and the axiom of empty set. Recall that the axiom schema of specification includes

\forall A\, \exist B\, \forall C\, (C \in B \Leftrightarrow [C \in A \and \theta(C)])

for each formula θ in the language of set theory in which B is not free.

The proof is as follows. Begin with a formula θ(C) that does not mention B, and a set A. If no element E of A satisfies θ then the set B desired by the relevant instance of the axiom schema of separation is the empty set. Otherwise, choose a fixed E in A such that θ(E) holds. Define a class function F such that F(D) = D if θ(D) holds and F(D) = E if θ(D) is false. Then the set B = F "A = A∩{x|θ(x)} exists, by the axiom of replacement, and is precisely the set B required for the axiom of specification.

This result shows that it is possible to axiomatize ZFC with a single infinite axiom schema. Because at least one such infinite schema is required (ZFC is not finitely axiomatizable), this shows that the axiom schema of replacement can stand as the only infinite axiom schema in ZFC if desired. Because the axiom schema of specification is not independent, it is sometimes omitted from contemporary statements of the Zermelo-Fraenkel axioms.

Specification is still important, however, for use in fragments of ZFC, because of historical considerations, and for comparison with alternative axiomatizations of set theory. A formulation of set theory that does not include the axiom of replacement will likely include some form of the axiom of specification, to ensure that its models contain a robust collection of sets. In the study of models of set theory, it is sometimes useful to consider models of ZFC without replacement.

The proof above uses the law of excluded middle in assuming that if A is nonempty then it must contain an element (in intuitionistic logic, a set is "empty" if it does not contain an element, and "nonempty" is the formal negation of this, which is weaker than "does contain an element"). The axiom of specification is included in intuitionistic set theory.

References