Computability and Logic

324 RAMSEY’S THEOREM
as follows. For any size-r subset x of ω, let p be its largest element. Then for any k
large enough that p < n + k, x will be in the domain of f k , and we have
f k (x) = f k+1 (x) = f k+2 (x) =···.
Let F(x) be this common value.
By the infinitary version of Ramsey’s theorem, F has an infinite homogeneous set.
That is, there is an infinite Y and a j with 1 ≤ j ≤ s such that F:[Y ] r →{j}. Let Z
be the set of the first n elements of Y , and take k large enough that the largest element
of Z is less than n + k. Then Z will be a size-n subset of {0,...,n + k − 1}, with
f k (x) = F(x) = j for all size-r subsets x of Z. In other words, Z will be a size-n
homogeneous set for f k , which is impossible, since f k is in T . This contradiction
completes the proof.
Proof of Lemma 26.3: To prove König’s lemma we are going to use the compactness
theorem. Let L T be the language with one one-place predicate B and with one
constant t for each node t in the tree T . Let Ɣ consist of the following quantifier-free
sentences:
(1)
Bs 1 ∨···∨Bs k
where s 1 , ..., s k are all the nodes in T 0 ;
(2)
∼(Bs & Bt)
for all pairs of nodes s, t belonging to the same level; and
(3)
∼Bs ∨ Bu 1 ∨···∨Bu m
for every node s, where u 1 , ..., u m are all the nodes immediately above s. [If there
are no nodes above s, the sentence (3) is just ∼Bs.]
We first show that if Ɣ has a model M, then T has an infinite branch. By (1) there
will be at least one node r in T 0 such that Br is true in M. By (2) there will in fact
be exactly one such node, call it r 0 . By (3) applied with r 0 as s, there will be at least
one node r immediately above r 0 such that Br is true in M. By (2) there will in fact be
exactly one such node, call it r 1 . Repeating the process, we obtain r 0 , r 1 , r 2 , ..., each
immediately above the one before, which is to say that we obtain an infinite branch.
We next show that Ɣ does have a model. By the compactness theorem, it is enough
to show that any finite subset of Ɣ has a model. In fact, we can show that for any
k, the set Ɣ k containing the sentence (1), all the sentences (2), and all the sentences
(3) for s of level less than k has a model. We can then, given a finite , apply this fact
to the least k such that all s occurring in are of level