K ¨onig’s Lemma, Compactness
First, for a little problem: There is a strange planet named Vlam on which the inhabitants are very much like us, except that they are immortal. The planet had a beginning in time, however. A curious thing about this planet is that, for any inhabitant x, if all children of x have blue eyes, so does x.
Problem 16.1. Suppose that an individual x on this planet has no chil-dren. Can it be determined from the above given condition whether x has blue eyes or not?
A more serious question: Can it be determined from the given condi-tion just what percentage of the inhabitants have blue eyes? Yes, it can, and the solution will emerge from the subject to which we now turn.
Generalized Induction
The principle of complete mathematical induction (Theorem 15.3), which is a theorem about the natural numbers, has an important generalization to arbitrary sets, even non-denumerable ones! We now consider an ar-bitrary set A of any size and a relation C(x, y) between elements of A, which we read “x is a component of y.” (The notion of component has many applications in set theory, number theory and logic. In set theory, the components of a set are the elements of the set. In some applications
173
174 III. Infinity
to number theory, m is said to be a component of n if m<n. In other applications, m is said to be a component of n if m+1=n. For logic, the notion of components of signed formulas (as defined in earlier chapters) will be seen to play a key role.)
Descending Chains
By a descending chain (for the component relation C(x, y)) is meant a finite sequence (x1, x2, . . . , xn) or a denumerable sequence (x1, x2, . . . , xn, . . .) such that each term of the sequence other than the first is a component of the preceding term (x2is a component of x1, x3is a component of x2, etc.).
We will be mainly concerned with component relations in which there are no infinite descending chains, because such relations will be seen to obey a very important induction principle. But, first:
Problem 16.2. Suppose that all descending chains are finite.
(a) Is it possible for an element x to be a component of itself?
(b) Is it possible for there to be two elements x and y such that x is a component of y and y is a component of x?
Problem 16.3.
(a) Consider the natural numbers 0, 1, 2, . . . , n, . . ., and define C(x, y) to be “y is the successor of x (i.e., x+1=y).” Are there any infinite descending chains?
(b) Suppose that instead of the natural numbers we consider the set of all whole numbers, positive and negative (. . . ,−3,−2,−1, 0, 1, 2, 3, 4, . . .). Are there any infinite descending chains?
Generalized Induction
We continue to consider a component relation C(x, y) on the elements of a set A. A property P of elements of A will be said to be inductive (with respect to the component relation, understood) if for every element x of A, if all components of x have the property P, so does x. This is understood to imply that if x has no components at all, then x must have the property, since it is vacuously true that all components of x (of which there are none) have the property. Again, I must remind the reader that if a set S has no elements at all, then anything we say about all elements of S must be true.
We now say that the component relation obeys the Generalized Induc-tion Principle if, for every inductive property P, P must hold for all ele-ments of A. Thus if the Generalized Induction Principle holds—that is,
16. Generalized Induction, K ¨onig’s Lemma, Compactness 175
if C(x, y)obeys the Generalized Induction Principle—then to show that a given property P holds for all elements of A, it suffices to show that for each element x of A, if all components of x have property P, so does x.
Let us note that the principle of complete mathematical induction (Theorem 15.3) is but a special case of generalized induction, in which A is the set of natural numbers and the component relation C(x, y) is x+1=y.
The following result is basic:
Theorem 16.1 (Generalized Induction Theorem). A sufficient condition for a component relation C(x, y)to obey the Generalized Induction Principle is that there be no infinite descending chains (all descending chains are finite). In other words, if all descending chains are finite, then C(x, y)obeys the General-ized Induction Principle.
Problem 16.4. Prove Theorem 16.1. Hint: If a property P is inductive, then for any element x of A, if P fails to hold for x, then it must fail for at least one component of x.
We can now answer the question we raised about the planet Vlam.
We take A to be the set of inhabitants of Vlam and the component rela-tion C(x, y)to be “x is a child of y.” Thus the components of x are the children of x. The given condition about blue eyes (x has blue eyes pro-vided all children of x do) is thus that the property of having blue eyes is inductive. A chain(x1, . . . , xn)now is simply a sequence in which for each i<n, xi+1is a child of xi. Obviously all such chains are finite, so, by Theorem 16.1, the Generalized Induction Principle holds, and since the property of having blue eyes is inductive, it follows that all inhabitants of Vlam have blue eyes!
Theorem 16.1 has a very important application to formulas of propo-sitional and first-order logic: We recall that we defined the components of an α to be α1and α2; those of β are β1and β2; those of γ are all formulas γ(a), where a is any parameter, and the components of δ are all formu-las δ(a). Obviously for any formula X, all components of X have fewer logical connectives or quantifiers than X, hence every descending chain starting with X must after finitely many steps eventuate in an atomic formula, so there are no infinite chains. Indeed, we define the degree of a formula as the number of occurrences of logical connectives or quan-tifiers, so no descending chain starting with a formula X can be longer than n, where n is the degree of X.
Since there are no infinite descending chains for formulas under our component relation, we have the following vital induction principles for formulas.
176 III. Infinity
Corollary 16.1 (Formula Induction). To show that a given property of formulas holds for all formulas, it suffices to show that, for any formula X, if the property holds for all components of X, then it holds for X too.
The converse of Theorem 16.1 also holds, namely:
Theorem 16.2. If the component relation obeys the generalized induction prin-ciple, then all descending chains are finite.
Problem 16.5. Prove Theorem T16.2. Hint: Consider the property P(x)
“all descending chains beginning with x are finite.”
Well-Foundedness
We continue to consider a component relation C(x, y)among elements of a set A. For any subset S of A, an element x of S is called an initial element of S if it has no components inside of S (either it has no components at all, or it does, but all of them lie outside of S).
Now we define the relation C(x, y) to be well-founded if every non-empty subset S of A contains at least one initial element.
As an example, the relation x+1=y on the natural numbers is well-founded (obviously for any non-empty set S of natural numbers, its least element is an initial element).
The following problem is easy.
Problem 16.6. Show that well-foundedness implies that there are no in-finite descending chains.
The following problem is a bit more tricky.
Problem 16.7. Prove that, if there are no infinite descending chains, then the component relation is well-founded.
By virtue of the last two problems and Theorems 16.1 and 16.2, we now have the following lovely result.
Theorem 16.3. For any component relation C(x, y)on a set A, the following three conditions are equivalent.
(1) The Generalized Induction Principle holds.
(2) There are no infinite descending chains.
(3) The relation is well-founded.
16. Generalized Induction, K ¨onig’s Lemma, Compactness 177