Metatheory - University of Cambridge

1. Induction 7
Case 3: n > 1 and n is not prime. Then n = ab with both n > a > 1
and n > b > 1. By assumption, a and b have prime factorisations,
i.e. a = p 1 × . . . × p j and b = q 1 × . . . × q k , for some prime numbers
p 1 , . . . , p j , q 1 , . . . , q k . So now n = p 1 × . . . × p j × q 1 × . . . × q k , i.e. n has
a prime factorisation.
So, in every case, n has the induction property on the assumption that every
smaller number has the induction property. Hence, by strong induction, every
number has the induction property.
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Note that we needed to use strong induction here, because in the case where
n = ab, we know nothing about a and b except that n > a > 1 and n > b > 1.
1.4 Induction on length
I now want to move on from arithmetical propositions to claims about TFL.
By the end of this book, we will have proved some rather sophisticated results.
But we need to start at the shallow end. In particular, we shall start by proving
a syntactic result, which should already be familiar:
Proposition 1.5. In any TFL sentence, there are exactly as many instances
of ‘(’ as instances of ‘)’.
When I mention ‘TFL sentences’ here I have in mind the the objects that
were defined recursively in fx C§6. In particular, then, I shall not allow us to
rely upon the convenient bracketing conventions that we introduced later in
fx C; so, we cannot, for example, allow for arbitrarily long conjunctions, and
we cannot drop the outermost brackets of a sentence. To save space, in what
follows I shall speak of ‘sentences’ rather than ‘TFL sentences as given by the
recursive definition of fx C§6’.
Now, we want to prove Proposition 1.5 by induction. But there is an immediate
obstacle in our path. The property mentioned in Proposition 1.5 is
x has as many instances of ‘(’ as of ‘)’, and that is a property of sentences,
rather than a property of numbers. So, if we are to prove this Proposition using
induction, we need a way to associate sentences with numbers.
The easiest way to associate sentences with numbers is just by considering
their length. More precisely, by the length of a sentence, we shall mean the
number of instances truth-functional connectives (i.e. ‘¬’, ‘∧, ‘∨’, ‘→’ and ‘↔’)
that the sentence contains. 5 Thus, for example:
• ‘B’ has length 0
• ‘A 234 ’ has length 0
• ‘¬(B ∨ C 4 )’ has length 2
• ‘¬¬B’ has length 2
• ‘¬¬¬B’ has length 3
Having defined the notion of the length of a sentence, we could now prove
Proposition 1.5 by (strong) induction on numbers, considered as lengths of
5 Many books refer to this as the complexity of a sentence.