Axiom Schema of Replacement - Example Applications

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.