You probably have equality symbol in the language as well. (This differs from one definition to the next.)Let L = {f } be a ﬁrst-order language containing a unary function

symbol f , and no other non-logical symbols.

1.Write down a sentence χ of L which is satisﬁable in some structure

with an inﬁnite domain but is false in every structure with a ﬁnite domain.

What can you say about the size of the domains of the models of the sentence

Describe f as a successor function in arithmetic. For 0 use the element that that does not have a predecessor (i.e., which is not a successor of anything). Then say that f is injective.