Computability and Logic

144 MODELS
the number of equivalence classes with exactly n elements. The examples to follow
are illustrated by pictures for equivalence relations of a variety of different signatures
in Figure 12-1.
(a) Signature (1, 0, 0, 0, 0, … )
(b) Signature (0, ∞, 0, 0, 0, … )
(c) Signature (0, 0, ∞, 0, 0, … )
…
…
…
(d)(i) Signatures (1, 1, 0, 0, 0, … ), (1, 2, 0, 0, 0, … ), (1, 3, 0, 0, 0, … ), and so on
…
…
…
(d)(ii) Signatures (0, ∞, 1, 0, 0, … ), (0, ∞, 0, 1, 0, …), (0, ∞, 0, 0, 1, … ) and so on
…
…
…
(d)(iii) Signature (1, ∞, 0, 0, 0, … )
…
…
(e) Signature (0, 0, 1, 0, 1, 0, 1, 0, … )
…
Figure 12-1. Equivalence relations.
12.7 Example (A promiscuous model). Let Ɣ a be the set containing Eq and the following
sentence E a :
∀x∀yx≡ y.
A denumerable model of Ɣ a consists of a denumerable set X with an equivalence relation
E in which all elements are in the same equivalence class, as in Figure 12-1(a). We claim
all such models are isomorphic. Indeed, if
X ={a 1 , a 2 , a 3 ,...} and Y ={b 1 , b 2 , b 3 ,...}
are any two denumerable sets, if X is the model with domain X and ≡ X the relation that
holds among all pairs a i , a j of elements of X, and if Y is the model with domain Y and ≡ Y
the relation that holds among all pairs b i , b j of elements of Y , then the function sending a i to