1

The movie titled ''The Madness of King George III'' was not a sequel to an earlier film called ''The Madness of King George II'', but we all know that not only kings, but movies too, sometimes use postnominal numerals in that way. And similarly with the subject line of this present question, since an earlier one that I posted is as follows:

Illustrative examples of a phenomenon in the logic of mathematical induction

Mathematical induction as taught to undergraduates almost always says one starts with a "base case."

There is a kind of mathematical induction that includes no base case, and that is this:

\begin{align} & \text{Prove that if for every natural number $m$ less than $n,\, P(m).$} \tag 1 \\ & \text{Conclude that for every natural number $n,\, P(n).$} \tag 2 \end{align}

No base case is needed because it is vacuously true that for every natural number $m$ less than the smallest natural number, $P(m),$ and so the base case is proved by line $(1)$ above.

So here is my question: Is there some variety of examples (or, at least, are there some examples) of problems that call for this form of induction and that are suitable for undergraduates being taught to understand what induction is and how it can be used? ("Suitable" might be taken as excluding theory of ordinal numbers in set theory since that would require an unduly large amount of exposition to those unfamiliar with it when a course must cover a fair number of other things.)

Michael Hardy
  • 1
  • To me, noting that the proposition is (vacuously) true for every number less than some given number is establishing the base case. I don't know if there's some formalism where there is a distinction between the two types, but in getting undergrads to understand proof by induction I'd expect to confuse more by presenting this alternative. – JonathanZ Mar 02 '22 at 20:10
  • @JonathanZsupportsMonicaC : Actually, the reason for the concept of vacuous truth is rarely explained well in teaching and exposition. That is one instance of the widespread intellectual sin of omitting to explain the reasons for particular definitions and concepts. – Michael Hardy Mar 02 '22 at 20:20
  • 1
    Unless I'm misreading, the form of induction you describe is called "strong induction" and is explicitly covered in many introductions to proof. – Mark S. Mar 03 '22 at 10:20
  • @MarkS. : "Strong induction" is not typically presented as a form in which no base case is needed. – Michael Hardy Mar 03 '22 at 15:41
  • @JonathanZsupportsMonicaC : But one need not make the base case a separate part of the argument in such instances. – Michael Hardy Mar 03 '22 at 15:42
  • 1
    The film was in fact called The Madness of King George allegedly because they did not want it to be thought to be a sequel, though it was based on a play called The Madness of George III. – Henry Apr 10 '22 at 08:22

1 Answers1

1

This kind of induction is often used in elementary number theory. Since in general there's no direct relation between multiplicative structure of $n-1$ and $n$, the "traditional" induction is often useless.

Instead, one may use induction about the smaller factors of a number to prove a property of a number.

Here's a proof that every natural number $n\ge 2$ is a product of primes.

If $n$ is a prime, we're done. If $n$ is not a prime, then there exists $a$, $b$ such that $n=ab$ with $1<a<n$ and $1<b<n$. Then, by the induction hypothesis, $a=p_1 p_2\cdots p_r$ and $b=q_1 q_2\cdots q_s$, then $n=ab = p_1 p_2\cdots p_r q_1 q_2\cdots q_s$

jjagmath
  • 18,214