Transfinite induction

Transfinite induction is an extension of mathematical induction to well-ordered sets, for instance to sets of ordinal numbers or cardinal numbers.

1 Transfinite induction
2 Transfinite recursion
3 Relationship to the axiom of choice
4 See also
5 Notes
6 References

Transfinite induction

Let P(α) be a property defined for all ordinals α. Suppose that whenever P(β) is true for all β < α, then P(α) is also true (including the case that P(0) is true given the vacuously true statement that P(α) is true for all $\alpha\in\emptyset$ ). Then transfinite induction tells us that P is true for all ordinals.

That is, if P(α) is true whenever P(β) is true for all β < α, then P(α) is true for all α. Or, more practically: in order to prove a property P for all ordinals α, one can assume that it is already known for all smaller β < α.

Usually the proof is broken down into three cases:

Zero case: Prove that P(0) is true.

Successor case: Prove that for any successor ordinal α+1, P(α+1) follows from P(α) (and, if necessary, P(β) for all β < α).

Limit case: Prove that for any limit ordinal λ, P(λ) follows from [P(β) for all β < λ].

Notice that all three cases are identical except for the type of ordinal considered. They do not formally need to be considered separately, but in practice the proofs are typically so different as to require separate presentations. Zero is sometimes considered a limit ordinal and then may sometimes be treated in proofs in the same case as limit ordinals.

Transfinite recursion

Transfinite recursion is a method of constructing or defining something and is closely related to the concept of transfinite induction. As an example, a sequence of sets A_α is defined for every ordinal α, by specifying how to determine A_α from the sequence of A_β for β < α.

More formally, we can state the Transfinite Recursion Theorem as follows. Given a class function G: V → V, there exists a unique transfinite sequence F: Ord → V (where Ord is the class of all ordinals) such that

F(α) = G(F $\upharpoonright$ α) for all ordinals α.

As in the case of induction, we may treat different types of ordinals separately: another formulation of transfinite recursion is that given a set g₁, and class functions G₂, G₃, there exists a unique function F: Ord → V such that

F(0) = g₁,
F(α + 1) = G₂(F(α)), for all α ∈ Ord,
F(λ) = G₃(F $\upharpoonright$ λ), for all limit λ ≠ 0.

Note that we require the domains of G₂, G₃ to be broad enough to make the above properties meaningful. The uniqueness of the sequence satisfying these properties can be proven using transfinite induction.

More generally, one can define objects by transfinite recursion on any well-founded relation R. (R need not even be a set; it can be a proper class, provided it is a set-like relation; that is, for any x, the collection of all y such that y R x must be a set.)

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.^[1] 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, giving a sequence $\langle r_{\alpha} | \alpha < c \rangle$ , where c is 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 choose 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.

Notes

^ In fact, the domain of the relation does not even need to be a set. It can be a proper class, provided that the relation R is set-like: for any x, the collection of all y such that y R x must be a set.

References

Patrick Suppes, 1972, Axiomatic set theory, Dover Publications. Section 7.1. ISBN 0-486-61630-4

Transfinite induction

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