Natural Numbers:Well-Ordering

Well-Ordering Principle

In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered by its "natural" or "magnitude" order in which $x$ precedes $y$ if and only if $y$ is either $x$ or the sum of $x$ and some positive integer (other orderings include the ordering $2,4,6,...$; and $1,3,5,...$).

The phrase "well-ordering principle" is sometimes taken to be synonymous with the "well-ordering theorem". On other occasions it is understood to be the proposition that the set of integers $\{...,-2,-1,0,1,2,3,...\}$ contains a well-ordered subset, called the natural numbers, in which every nonempty subset contains a least element.

Depending on the framework in which the natural numbers are introduced, this (second order) property of the set of natural numbers is either an axiom or a provable theorem. For example:

In Peano arithmetic, second-order arithmetic and related systems, and indeed in most (not necessarily formal) mathematical treatments of the well-ordering principle, the principle is derived from the principle of mathematical induction, which is itself taken as basic. Considering the natural numbers as a subset of the real numbers, and assuming that we know already that the real numbers are complete (again, either as an axiom or a theorem about the real number system), i.e., every bounded (from below) set has an infimum, then also every set $A$ of natural numbers has an infimum, say $a^{*}a^{*}$. We can now find an integer $n^{*}n^{*}$ such that $a^{*}a^{*}$ lies in the half-open interval $(n^{*}-1,n^{*}]$, and can then show that we must have $a^{*}=n^{*}$, and $n^{*}n^{*}$ in $A$. In axiomatic set theory, the natural numbers are defined as the smallest inductive set (i.e., set containing 0 and closed under the successor operation). One can (even without invoking the regularity axiom) show that the set of all natural numbers $n$ such that "$\{0,\ldots ,n\}$ is well-ordered" is inductive, and must therefore contain all natural numbers; from this property one can conclude that the set of all natural numbers is also well-ordered. In the second sense, this phrase is used when that proposition is relied on for the purpose of justifying proofs that take the following form: to prove that every natural number belongs to a specified set $S$, assume the contrary, which implies that the set of counterexamples is non-empty and thus contains a smallest counterexample. Then show that for any counterexample there is a still smaller counterexample, producing a contradiction. This mode of argument is the contrapositive of proof by complete induction. It is known light-heartedly as the "minimal criminal" method and is similar in its nature to Fermat's method of "infinite descent".

Representation of the ordinal numbers up to $\omega ^{\omega }$. Each turn of the spiral represents one power of $\omega$. Transfinite induction requires proving a base case (used for 0), a successor case (used for those ordinals which have a predecessor), and a limit case (used for ordinals which don't have a predecessor).

Transfinite induction

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

Induction by cases

Let $P(\alpha )$ be a property defined for all ordinals $\alpha$. Suppose that whenever $P(\beta )$ is true for all $\beta <\alpha$, then $P(\alpha )$ is also true. It is not necessary here to assume separately that $P(0)$ is true. As there is no $\beta$ less than 0, it is vacuously true that for all $\beta <0$, $P(\beta )$ is true. Then transfinite induction tells us that $P$ is true for all ordinals.

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 $\alpha +1$, $P(\alpha +1)$ follows from $P(\alpha )$ (and, if necessary, $P(\beta )$ for all $\beta <\alpha$).
• Limit case: Prove that for any limit ordinal $\lambda$, $P(\lambda )$ follows from $P(\beta )$ for all $\beta <\lambda$.

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 similar to transfinite induction; however, instead of proving that something holds for all ordinal numbers, we construct a sequence of objects, one for each ordinal.

As an example, a basis for a (possibly infinite-dimensional) vector space can be created by choosing a vector $v_{0}$ and for each ordinal α choosing a vector that is not in the span of the vectors $\{v_{\beta }\mid \beta <\alpha \}$. This process stops when no vector can be chosen.

More formally, we can state the Transfinite Recursion Theorem as follows:

• Transfinite Recursion Theorem (version 1). Given a class function $G:V\to V$ (where $V$ is the class of all sets), there exists a unique transfinite sequence $F:{\text{Ord}}\to V$ (where Ord is the class of all ordinals) such that

F(α) = G(F $\upharpoonright$ α) for all ordinals α, where $\upharpoonright$ denotes the restriction of F's domain to ordinals < α.

As in the case of induction, we may treat different types of ordinals separately: another formulation of transfinite recursion is the following:

• Transfinite Recursion Theorem (version 2). 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 proved 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; i.e. for any x, the collection of all y such that yRx is 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. 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. 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 $\langle r_{\alpha }|\alpha <\beta \rangle$, 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.

Licensing

Content obtained and/or adapted from:

• Well-ordering principle, Wikipedia under a CC BY-SA license
• Transfinite induction, Wikipedia under a CC BY-SA license