3 Mathematical Induction Reading: Metalogic Part II, 23,27 Contents 3.1 Mathematical Induction . 35 3.1.1 Peano's Axioms . 35 3.1.2 **Example: Parentheses . 39 3.1.3 The Principle of Induction on the Construction ofaW®.................................... 40 3.1.4 **The Induction Principle . 41 3.2 Unique Decomposition . 43 3.2.1 Unique Readability Theorem . 43 3.2.2 Finding the Main Connnective . 45 3.3 Some Theorems Involving Induction . 46 3.3.1 Unique Valuation . 46 3.3.2 The Interpolation Theorem . 46 3.3.3 Disjunctive Normal Form . 47 3.1 Mathematical Induction Our objective in this section is to familiarize the reader with the technique of Proof by Mathematical Induction. This proof technique is used throughout mathematical logic, and so we should learn it right away. 3.1.1 Peano's Axioms We think of the natural numbers as forming a sequence beginning with 0 followed by its successor, and then its successor, and so on. We begin with 0 and then generate the sequence by adding on n0 whenever we have generated n. n0 is said to be the successor of n, and it may be thought of as n + 1. We thus get the sequence 0; 00; 000; 0000; 00000;::: 36 3 Mathematical Induction This informal characterization of the set of natural numbers N, is formally characterized by Peano's Postulates1 P1 0 2 N. 0 P2 If n 2 N, then there is a unique n 2 N. 0 0 P3 For all m; n 2 N, if m = n , then m = n. 0 P4 For each n 2 N, n 6= 0. 0 P5 If M ⊆ N such that 0 2 M and m 2 M whenever m 2 M, then M = N. 0 P1 is straightforward. P2 tells us that is a function, so that if x = y, then 0 0 0 x = y .P3 tells us that the function is 1-1, i.e., that no number can be the successor of two distinct numbers. P4 tells us that 0 is not the successor of any number, so that, in e®ect, the Range of the function 0 is the set N ¡ f0g. And P5 is the basis for what is known as Weak Induction Principle 3.1.1 (The Principle of Mathematical Induction) Let P be a property of natural numbers. Suppose that: (1) 0 has the property P. (2) If any natural number n has the property P, then its successor n0 has the property P. Then, every natural number n has the property P. The Principle of Mathematical Induction can be drawn from P5 by letting M = fm 2 NjP (m)g. The following terminology is fairly standard in conjunction with proof by mathematical induction. The part of the proof which consists in establishing (1), i.e., that 0 has P, is called the Basis step. The part of the proof which consists in establishing (2), i.e., that n0 has the property P whenever n has the property P, is called the Induction Step. The induction step is carried out by assuming that n has P, and showing that on this assumption n0 has P. This assumption that n has P is called the Induction Hypothesis. We will use mathematical induction to prove the following elementary consequence of these postulates, namely, that every natural number other than 0 is the successor of a natural number2: Proposition 20 Every n 2 N is such that either n = 0 or else there exists an m 2 N and n = m0 Proof By Mathematical Induction. Basis Step. n = 0. The disjunction follows immediately by addition: either n = 0 or else there exists an m 2 N and n = m0. Induction Step. Assume the theorem holds for n 2 N, i.e., assume that either n = 0 or else there exists an m 2 N and n = m0. (This is the Induction 1 These are actually due to Dedekind 2 This establishes that the function 0 is not merely 1-1, but also onto N ¡ f0g 3.1 Mathematical Induction 37 Hypothesis.) We have to show that this also holds for n0. Clearly n0 2 N, 0 by P2. And, clearly n 6= 0, by P4. So, we have to show that there exists an m 2 N such that n0 = m0. But, of course, there is: let m = n. ¤ There are a number of variants of this principle that mathematicians rely on in proofs by induction. Kleene (1950) also acknowledges Course-of-Values Induction, on which the induction hypothesis is not simply that n has P, but that m has P, for all m · n. So, we have Principle 3.1.2 (The Principle of Course-of-Values Induction:) Let P be a property of natural numbers. Suppose that: (1) 0 has the property P. (2) If all natural numbers m · n have the property P, then n0 has the property P. Then, every natural number n has the property P. [Andrews, Mendelson (1987) identi¯es a Principle of Complete Induction,3 This is also known as Strong Induction: Principle 3.1.3 (The Principle of Complete Induction (Strong Induction):) Let P be a property of natural numbers. Suppose that: If all natural numbers m < n have the property P, then n has the property P. Then, every natural number n has the property P. These variations need not upset us, however, for they are provably equiv- alent. As such, we can choose whichever form makes the proof easiest. We will show that the Principle of Mathematical Induction is equivalent to the Principle of Complete Induction in two steps. Lemma 3.1.1 The Principle of Mathematical Induction is a consequence of The Principle of Complete Induction. Proof [Andrews] We want to show that if P (0) and if, for all n, if P (n) then P (n0), then for all n, P (n). So, (as with any conditional proof), we assume the antecedent and try to derive the consequent. So, we assume (*) P (0) and for all n, if P (n) then P (n0) We want to show that P (n) for all n. We do so using the Principle of Complete Induction, i.e., we take as our induction hypothesis that P (j) for all j < n, and try to derive P (n). The Principle of Complete Induction will then entitle 3 Hunter (1996) has a di®erent terminology. \In the Induction Step we prove that if the Theorem holds for all cases up to an arbitrarily given point, then it holds also for all cases at the next higher point. (This will be a strong induction. A weak induction is one in which the Induction Step shows that if the Theorem holds for all cases at an arbitrarily given point, then it holds also for all cases at the next higher point." By these lights, our Principle of Mathematical Induction would be a weak induction and both our Course-of-Values Induction and our Principle of Complete Induction would be strong inductions|assuming, of course, that Hunter would count the Induction Hypothesis in Complete Induction as the Induction Step even though there is no Basis Step. 38 3 Mathematical Induction us to conclude that P (n) for all n. Induction Step. Assume P (j) for all j < n. If n = 0, then by (*), P (n). Otherwise n > 0. So, n = m0. In that case, m < n, so, P (m) by the induction hypothesis. And by (*), for every number k, if P (k) then P (k0), so P (m0). So, for all n, P (n). ¤ Lemma 3.1.2 The Principle of Complete Induction is a consequence of The Principle of Mathematical Induction Proof [Andrews] We want to show that if, for every number n, P (n) whenever all number j < n are such that P (j), then P (n) for all n. So, we make the following assumption: (**) For all n, P (n) whenever all number j < n are such that P (j) We must show that P (n) for all n. We prove this using The Principle of Mathematical Induction. Basis Step. n = 0. Since there are no j < 0, then by (**) it is trivial that P (0). Induction Step. The Induction hypothesis is that P (n). We want to show P (n0). But, by (**), since P (n), then for all j < n, P (j). So, since P (n) and for all j < n, P (j), it follows that, for all k < n0, P (k). By (**), P (n0). So, for all n, P (n). QED From these two lemmas, we derive Lemma 3.1.3 The Principle of Mathematical Induction is equivalent to the Principle of Complete Induction Problems 3.1. The Principle of Mathematical Induction is a consequence of Course-of- Values Induction. 3.2. Course-of-Values Induction is a consequence of The Principle of Mathe- matical Induction. 3.3. The Principle of Complete Induction is a consequence of Course- of- Values Induction. 3.4. Course-of-Values Induction is a consequence of The Principle of Complete Induction. 3.5. Suppose that the Basis step were eliminated from Course-of Values In- duction to get: Let P be a property of natural numbers. Suppose that: If all natural numbers m · n have the property P, then n0 has the property P. Then, every natural number n has the property P. Show that this is equivalent to The Principle of Mathematical Induction. 3.1 Mathematical Induction 39 3.1.2 **Example: Parentheses We take a very liberal attitude about what is to count as a property of num- bers.