Transfinite Induction - Relationship To The Axiom of Choice

Relationship To The Axiom of Choice

Proofs or constructions using induction and recursion often use the axiom of choice to produce a well-ordered relation that can be treated by transfinite induction. However, if the relation in question is already well-ordered, one can often use transfinite induction without invoking the axiom of choice. For example, many results about Borel sets are proved by transfinite induction on the ordinal rank of the set; these ranks are already well-ordered, so the axiom of choice is not needed to well-order them.

The following construction of the Vitali set shows one way that the axiom of choice can be used in a proof by transfinite induction:

First, well-order the real numbers (this is where the axiom of choice enters via the well-ordering theorem), giving a sequence, where β is an ordinal with the cardinality of the continuum. Let v₀ equal r₀. Then let v₁ equal r_α1, where α₁ is least such that r_α1 − v₀ is not a rational number. Continue; at each step use the least real from the r sequence that does not have a rational difference with any element thus far constructed in the v sequence. Continue until all the reals in the r sequence are exhausted. The final v sequence will enumerate the Vitali set.

The above argument uses the axiom of choice in an essential way at the very beginning, in order to well-order the reals. After that step, the axiom of choice is not used again.

Other uses of the axiom of choice are more subtle. For example, a construction by transfinite recursion frequently will not specify a unique value for A_α+1, given the sequence up to α, but will specify only a condition that A_α+1 must satisfy, and argue that there is at least one set satisfying this condition. If it is not possible to define a unique example of such a set at each stage, then it may be necessary to invoke (some form of) the axiom of choice to select one such at each step. For inductions and recursions of countable length, the weaker axiom of dependent choice is sufficient. Because there are models of Zermelo–Fraenkel set theory of interest to set theorists that satisfy the axiom of dependent choice but not the full axiom of choice, the knowledge that a particular proof only requires dependent choice can be useful.