Computability and Logic

26.2. K ÖNIG’S LEMMA 325
The presence of t 0 in the denotation of B will make (1) true. Since we have included
in the denotation of B only one node t i at each level T i for i ≤ k and none at higher
levels, (2) will be true. For a sentence of form (3) with s of level i < k, the first disjunct
will be true unless s is t i , in which case the Bt i+1 will be among the other disjuncts,
and will be true. In either case, then, (3) will be true for every sentence of this type in
Ɣ k . [The sentence of form (3) with t k as s will be false, but that sentence is not in Ɣ k .]
Before indicating the connection of Ramsey’s theorem with the kind of logical
phenomena we have been concerned with in this book, we digress a moment to
present a pretty application of Ramsey’s theorem.
26.4 Corollary (Schur’s theorem). Suppose that each natural number is ‘painted’ exactly
one of some finite number of ‘colors’. Then there are positive integers x, y, z all the
same color such that x + y = z.
Proof: Suppose the number of colors is s. Paint each size-2 set {i, j}, i < j, the
same color as the natural number j − i. By Ramsey’s theorem (r = 2, n = 3), there
are a positive integer m ≥ 3 and a size-3 subset {i, j, k} of {0, 1,...,m − 1} with
i < j < k, such that {i, j}, { j, k} and {i, k} are all the same color. Let x = j − i,
y = k − j, and z = k − i. Then x, y, z are positive integers all the same color, and
x + y = z.
Ramsey’s theorem is, in fact, just the starting point for a large body of results in
combinatorial mathematics. It is possible to add some bells and whistles to the basic
statement of the theorem. Call a nonempty set Y of natural numbers glorious if Y
has more than p elements, where p is the least element of Y . Since every infinite set
is automatically glorious, it would add nothing to the infinitary version of Ramsey’s
theorem to change ‘infinite homogeneous set’ to ‘glorious infinite homogeneous set’.
It does, however, add something to the statement of the original Ramsey’s theorem
to change ‘size-n homogeneous set’ to ‘glorious size-n homogeneous set’.
Let us call the result of this change the glorified Ramsey’s theorem. Essentially
the same proof we have given for Ramsey’s theorem proves the glorified Ramsey’s
theorem. (At the beginning, take T to be the set of partitions without glorious size-n
homogeneous sets, and towards the end, take Z to be the set of the first q elements
of Y , where q is the maximum of n and p, p being the least element of Y.) There is,
however, an interesting difference in logical status between the two.
While the proof we have presented for Ramsey’s theorem involved a detour through
the infinite, F. P. Ramsey’s original proof of Ramsey’s theorem did not. Using a reasonable
coding of finite sets of natural numbers by natural numbers, Ramsey’s theorem
can be expressed in the language of arithmetic, and by ‘formalizing’ Ramsey’s proof,
it can be proved in P. By contrast, the glorified Ramsey’s theorem, though it can be
expressed in the language of arithmetic, cannot be proved in P.
It is an example of a sentence undecidable in P that is far more natural, mathematically
speaking, than any we have encountered so far. (The sentences involved in
Gödel’s theorem or Chaitin’s theorem, for instance, are ‘metamathematical’, being
about provability and computability, not ordinary mathematical notions on the order of
those occurring in Ramsey’s theorem.) Unfortunately, the Paris–Harrington theorem,