Computability and Logic

128 THE UNDECIDABILITY OF FIRST-ORDER LOGIC
The ‘background information’ is provided by the following:
(1)
(2)
(3)
∀u∀v∀w(((Suv & Suw) → v = w)&((Svu & Swu) → v = w))
∀u∀v(Suv → u < v) &∀u∀v∀w((u < v & v < w) → u < w)
∀u∀v(u < v → u ≠ v).
These say that a number has only one successor and only one predecessor, that a
number is less than its predecessor, and so on, and are all equally true in the standard
interpretation.
It will be convenient to introduce abbreviations for the mth-successor relation,
writing
S 0 uv for u = v
S 1 uv for Suv
S 2 uv for ∃y(Suy & Syv)
S 3 uv for ∃y 1 ∃y 2 (Suy 1 & Sy 1 y 2 & Sy 2 v)
and so on. (In S 2 , y may be any convenient variable distinct from u and v; for
definiteness let us say the first on our official list of variables. Similarly for S 3 .) The
following are then true in the standard interpretation.
(4)
(5)
(6)
(7)
(8)
∀u∀v∀w(((S m uv & S m uw) → v = w)&((S m vu & S m wu) → v = w))
∀u∀v(S m uv → u < v) if m ≠ 0
∀u∀v(S m uv → u ≠ v) if m ≠ 0
∀u∀v∀w((S m wu & Suv) → S k wv) if k = m + 1
∀u∀v∀w((S k wv & Suv) → S m wu) if m = k − 1.
Indeed, these are logical consequences of (1)–(3) and hence of Ɣ, true in any interpretation
where Ɣ is true: (4) follows on repeated application of (1); (5) follows
on repeated application of (2); (6) follows from (3) and (5); (7) is immediate from
the definitions; and (8) follows from (7) and (1). If we also write S −m uv for S m vu,
(4)–(8) still hold.
We need some further notational conventions before writing down the remaining
sentences of Ɣ. Though officially our language contains only the numeral 0 and
not numerals 1, 2, 3, or−1, −2, −3, it will be suggestive to write y = 1, y = 2,
y = −1, and the like for S 1 (0, y), S 2 (0, y), S −1 (0, y), and so on, and to understand
the application of a predicate to a numeral in the natural way, so that, for instance,
Q i 2 and S2u abbreviate ∃y(y = 2 &Q i y) and ∃y(y = 2 & Syu). A little thought
shows that with these conventions (6)–(8) above (applied with 0 for w) giveusthe
following wherein p, q, and so on, are the numerals for the numbers p, q, and so
on:
(9)
(10)
(11)
p ≠ q if p ≠ q
∀v(Smv → v = k) where k = m + 1
∀u(Suk → u = m) where m = k − 1.