Metatheory - University of Cambridge

1. Induction 4
knocking the first one down. To knock down the first domino corresponds with
proving that 0 has ϕ. And to arrange the dominoes in this way corresponds
with proving the conditional: if n has ϕ then n + 1 has ϕ.
Of course, this is just an analogy, and a spatial one at that. Why, then,
should we believe the Principle of Induction? Here is an argument in its favour.
Suppose you have proved both that 0 has ϕ and also that, for every n, if n has
ϕ then n + 1 has ϕ. Now, pick any number, m. Given what you have proved,
you would be justified in writing down all of the following:
0 has ϕ
if 0 has ϕ, then 1 has ϕ
if 1 has ϕ, then 2 has ϕ
.
if m − 1 has ϕ, then m has ϕ
It would now follow, by m-many applications of modus ponens, that m has ϕ.
And since m was arbitrary, this point generalises: every number has ϕ.
I doubt this line of argument will convince the determined sceptic. 2 And,
in the end, the question ‘Why believe in the Principle of Induction?’ is a deep
question in the philosophy of mathematics. That question is interesting, but it
is a question for another day. For the purposes of this book, we shall take the
Principle of Induction for granted.
1.2 Arithmetical proofs involving induction
To invoke induction in a proof, we must first identify some property, ϕ, that
concerns us. We call this the induction property. We then do two things:
• Prove the base case: we need to show that 0 has ϕ. Often, this is a fairly
straightforward task.
• Prove the induction case: we need to show that, for any n, if n has ϕ, so
does n + 1. This is normally the harder step. We typically proceed by
‘choosing some arbitrary n’, and assuming ‘for induction’ that n has ϕ.
If, on just this basis, we can show that n + 1 has ϕ, then we have proved
the conditional: if n has ϕ, then n + 1 has ϕ. And since n was arbitrary,
this holds for all n.
To illustrate the strategy, I shall offer a few simple examples of the use of
induction, in establishing some arithmetical results. (Just to be clear: I am
not going to make any use of the arithmetical results themselves; I just want
to show how induction is used in proofs.) So, for example, suppose we want to
prove a claim that should be very obvious:
Proposition 1.1. Every number is either even or odd. That is: for every
number n, either there is a number k such that n = 2k (i.e. n is even), or there
is a number k such that n = 2k + 1 (i.e. n is odd).
I shall prove this Proposition by induction. The induction property will be x
is either odd or even. It suffices to prove the base case and the induction case;
then all we need to complete the proof is to appeal to induction. Here goes:
2 It is worth spending some time asking yourself: Why won’t this convince the sceptic?