Computability and Logic

25.1. ORDER IN NONSTANDARD MODELS 303
THAN a third, then the first is LESS THAN the third. LESS THAN is a linear ordering of
the NUMBERS, just as less than is a linear ordering of the numbers.
Zero is the least number, so ZERO is the LEAST NUMBER. Any number is less than
its successor, and there is no number between a given number and its successor (in
the sense of being greater than the former and less than the latter), so any NUMBER
is LESS THAN its SUCCESSOR, and there is no NUMBER between a given NUMBER and
its SUCCESSOR. In particular, 0 ′ (that is, 1 or one) is the next-to-least number, and O †
(which we may call I or ONE) is the next-to-LEAST NUMBER;0 ′′ is next-to-next-to-least
and O †† is next-to-next-to-LEAST; and so on. So there is an initial segment O, O † ,
O †† , ...of the relation LESS THAN that is isomorphic to the series 0, 0 ′′ , 0 ′′ , ...of the
(natural) numbers.
We call O, O † , O †† ,...the standard NUMBERS. Any others are nonstandard. The
standard NUMBERS are precisely those that can be obtained from ZERO by applying
the SUCCESSOR operation a finite number of times. For any (natural) number n, let us
write h(n) for O ††...†(n times) ′′... ′(n times)
, which is the denotation of the numeral n or 0
in M. Then the standard NUMBERS are precisely the h(n) for n a natural number. Any
others are nonstandard. Any standard NUMBER h(n) isLESS THAN any nonstandard
NUMBER m. This is because, being true in N , the sentence
∀z((z ≠ 0 & ... & z ≠ n) → n < z)
must be true in M,soanyNUMBER other than h(0),...,h(n) must be GREATER THAN
h(n).
[It is not quite trivial to show that there must be some nonstandard NUMBERS
in any nonstandard model M. If there were not, then h would be a function from
(natural) numbers onto the domain of M. We claim that in that case, h would be
an isomorphism between N and M, which it cannot be if M is nonstandard. First,
h would be one-to-one, because when m ≠ n, m ≠ n is true in N and so in M, so
the denotations of m and n in M are distinct, that is, h(m) ≠ h(n). Further, when
m + n = p, m + n = p is true in N and so in M, soh(m + n) = h(m) ⊕ h(n).
Finally, h(m · n) = h(m) ⊗ h(n) by a similar argument.]
Any number other than zero is the successor of some unique NUMBER, soany
NUMBER other than ZERO is the SUCCESSOR of some unique number. So we can define
a function ‡ from NUMBERS to NUMBERS by letting O ‡ = O and otherwise letting m ‡
be the unique NUMBER of which m is the SUCCESSOR.Ifn is standard, then n † and n ‡
are standard, too, and if m is nonstandard, then m † and m ‡ are nonstandard. Moreover,
if n is standard and m nonstandard, then n is LESS THAN m ‡ .
We’ll now define an equivalence relation ≈ on NUMBERS.Ifa and b are NUMBERS,
we’ll say that a ≈ b if for some standard(!) NUMBER c, either a ⊕ c = b or b ⊕ c = a.
Intuitively speaking, a ≈ b if a and b are a finite distance away from each other, or
in other words, if one can get from a to b by applying † or ‡ a finite number of times.
Every standard NUMBER bears the relation ≈ to all and only the standard NUMBERS.
We call the equivalence class under ≈ of any NUMBER a the block of a. Thus a’s
block is
{...,a ‡‡‡ , a ‡‡ , a ‡ , a, a † , a †† , a ††† ,...}.