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The way I understand complete induction, as applied to the naturals at least, the inductive step consists of assuming that a given proposition $p_i$ is true for $1 \le i \le n$, and from this deduce the truth of of $p_{n+1}$. However, I had thought that one always needed to check the base case ($i=1$). But reading the articles on vacuous truth and transfinite induction, I am now left wandering if this is really the case, considering that the truth of $p_1$ might follow (vacuously) from the fact that there is no smaller element. So my two questions are:

  • When proving some statement with complete induction, is it correct to assume the truth of $p_1$ as a vacuous truth? Could anyone provide an example? (somewhat to my astonishment I was unable to find one...)

  • If $p_1$ is vacuously true for complete induction, then is that also the case for (simple) induction? I am inclined to believe so, because the truth of $p_1$ would follow vacuously from the truth of (the set containing) the previous smaller element, which is the empty set. However in all I've read of simple induction, this never comes up; instead one is always told to check the base case.

Thank you in advance for your help in clarifying this!

wmnorth
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2 Answers2

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See the answer to this post.

Mathematical induction consists into applying the following "method of proof", based on axiom schema of induction :

if $P(0)$ and $\forall k [P(k) \implies P(k+1)]$, then $\forall n P(n)$.

In words, I have to prove that : property $P$ hold for number $0$ and that : if property $P$ holds for a number $k$ whatever, then it holds also for its successor $k+1$, in order to be able to conclude that : $P$ holds for all natural numbers.

Strong induction is the equivalent form :

if $\forall i [(i < k \implies P(i)) \implies P(k)]$, then $\forall n P(n)$.

In words, I've to prove that : if $P(i)$ holds for all $i < k$, then $P(k)$ holds, in order to be able to conclude that : $P$ holds for all natural numbers.

Comment

I'm starting from $0$ but we can start from $1$; it depends on the definition of natural number.

We can start from a number $k > 0$, if we want to prove a statement not for all natural numbers but only for all numbers greater than or equal to $k$. In this case we have :

  • to prove that the statement holds when $n = k$;

  • to prove that if the statement holds for $n = m > k$ then the same statement also holds for $n = m$.

In every case, the proof will not work if $P$ does not hold for the basis case : $P(0)$ (or $P(k)$).

For math induction, we have to prove it as first step (the basis case), while for strong induction it is included into the proof of $\forall i [(i < k \implies P(i)) \implies P(k)]$.

Community
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Mauro ALLEGRANZA
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"Theorem": for every positive integer $n$, we have $n<n-1$.

"Proof": By "induction". Suppose that $k<k-1$. Adding one to both sides, we get $$k+1 <(k-1)+1=(k+1)-1.$$ By the Induction Principle, $n<n-1$ for all $n$. QED.

So there you have a "proof" that $2<1$.

"Vacuosly True" in this case refers to the following. When you do complete induction (transfinite or not), the base case is included in the assertion if $p(k)$ for all $k<n$, then $p(n)$. Because there is a quantifier, when $0$ is the first element $p(k)$ for all $k<0$ is true; so to prove the implication for $n=0$ amounts to prove $p(0)$.

Martin Argerami
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