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"?