Computability and Logic

12.2. EQUIVALENCE RELATIONS 145
b i is an isomorphism between X and Y. Condition (I1) in the definition of isomorphism—
the only applicable condition—says that we must in all cases have a i ≡ X a j if and only if
f (a i ) ≡ Y f (a j ); and of course we do, since we always have both a i ≡ X a j and b i ≡ Y b j .
Thus Ɣ a has only one isomorphism type of denumerable model.
12.8 Example (An eremitic model). Let Ɣ b be the set containing Eq and the following
sentence E b :
∀x∀y (x ≡ y ↔ x = y).
A denumerable model of Ɣ b consists of a denumerable set X with an equivalence relation
E in which each element is equivalent only to itself, so each equivalence class consists
of but a single element, as in Figure 12-1(b). Again any two such models are isomorphic.
With the notation as in the preceding example, this time we have a i ≡ X a j if and only if
f (a i ) ≡ Y f (a j ), because we only have a i ≡ X a j when i = j, which is precisely when we
have b i ≡ Y b j .
12.9 Example (Two isomorphism types). Let Ɣ ab be the set containing Eq and the disjunction
E a ∨ E b . Any model of Ɣ ab must be either a model of Ɣ a or one of Ɣ b , and all
models of either are models of Ɣ ab . Now all denumerable models of Ɣ a are isomorphic to
each other, and all denumerable models of Ɣ b are isomorphic to each other. But a model of
Ɣ a cannot be isomorphic to a model of Ɣ b , by the isomorphism lemma, since E a is true in
the former and false in the latter, and inversely for E b .SoƔ ab has exactly two isomorphism
types of denumerable model.
12.10 Example (An uxorious model). Let Ɣ c be the set containing Eq and the following
sentence E c :
∀x∃y(x ≠ y & x ≡ y & ∀z(z ≡ x → (z = x ∨ z = y))).
A denumerable model of Ɣ c consists of a denumerable set X with an equivalence relation
E in which each element is equivalent to just one other element than itself, so each equivalence
class consists of exactly two elements, as in Figure 12-1(c). Again there is only one
isomorphism type of denumerable model. If we renumber the elements of X so that a 2 is
the equivalent of a 1 , a 4 of a 3 , and so on, and if we similarly renumber the elements of Y ,
again the function f (a i ) = b i will be an isomorphism.
12.11 Example (Three isomorphism types). Let Ɣ abc be the set containing Eq and the
disjunction E a ∨ E b ∨ E c . Then Ɣ abc has three isomorphism types of denumerable models.
The reader will see the pattern emerging: we can get an example with n isomorphism types
of denumerable models for any positive integer n.
12.12 Example (Denumerably many isomorphism types). Let Ɣ d be the set containing Eq
and the following sentence E d :
∀x∀y((∃u(u ≠ x & u ≡ x)&∃v(v ≠ y & v ≡ y)) → x ≡ y).
A denumerable model of Ɣ d will consist of a denumerable set X with an equivalence
relation in which any two elements a and b that are not isolated, that is, that are such that
each is equivalent to something other than itself, are equivalent to each other. Here there
are a number of possible pictures. It could be that all elements are equivalent, or that all