First, say that all 's are distinct. Second, that they are the whole universe, i.e., any other is equal to one of 's. Finally, for each pair say whether holds on .
Consider a language with equality whose only parameter (aside from ) is a two-place predicate symbol . Show that if is finite and ( is elementary equivalent to ), is isomorphic to .
Suggestion: Suppose the universe of has size . Make a single sentence of the form that describes "completely". That is, on the one hand, must be true in . And on the other hand, any model of must be exactly like (i.e., isomorphic to) .
To get started, how do I make a single sentence of the form that describes "completely"?
Let be the binary relation to which two place predicate symbol is assigned by .
implies that if , then . Therefore, the cardinality of B is the same with A, since ought to be true in its model B by assumption.
Without loss of generality, , and .
The required isomorphism is given by for satisfying
Is this correct?
is easy to prove (for example, because is symmetric and holds), and so I only need to show .
Let's denote . We have and , i.e., and for some and . Then iff , so h such that for is an isomorphism.