Computability and Logic

21.3. DYADIC LOGIC 277
We then claim S is satisfiable if and only if S* is satisfiable. As in the preceding
proofs, the ‘if’ direction is easy, and for the ‘only if’ direction what we need to do
is to show, given a model M of S, which may be taken to have domain the natural
numbers, that we can define an interpretation M*, also with domain the natural
numbers, which will assign as denotation to Q in M* a relation such that for any
natural numbers b 1 , b 2 , b 3 , those numbers will satisfy P ∗ (x 1 , x 2 , x 3 )inM* ifand
only if those numbers satisfy P(x 1 , x 2 , x 3 )inM. To accomplish this last and so
complete the proof, it will be enough to establish the following lemma.
21.13 Lemma. Let R be a three-place relation on the natural numbers. Then there is
a two-place relation S on the natural numbers such that if a, b, c are any natural numbers,
then we have Rabc if and only if for some natural numbers w, x, y, z we have
(1)
∼Sww & Swx & Sxy & Syz & Szw &
Swa & Sxb& Syc& ∼Sax & ∼Sby & ∼Scz & Sza.
Proof: One of the several ways of enumerating all triples of natural numbers is to
order them by their sums, and where these are the same by their first components,
and where these also are the same by their second components, and where these also
are the same by their third components. Thus the first few triples are
(0, 0, 0)
(0, 0, 1)
(0, 1, 0)
(1, 0, 0)
(0, 0, 2)
(0, 1, 1)
(0, 2, 0)
(1, 0, 1)
(1, 1, 0)
(2, 0, 0)
(0, 0, 3)
.
·
Counting the initial triple as the first rather than the zeroth, it is clear that if the nth
triple is (a, b, c), then a, b, c are all