Meaning of finite (and hence infinite) model in mathematical logic

Question

In the book by enderton, A mathematical introduction to logic. Phases or terms that involves, "infinite models" and "finite models" appears, especially in section 2.6.

Example:

"Some sentences have only infinite models, like the sentence saying $<$ is an ordering with no largest element"

"It is also possible to have sentences having only finite models........"

So I'm not exactly sure what is the meaning of "finite/infinite models", hence I am if someone could explain to me, preferably with examples to illustrate.

Thank you in advance .

Answer 1

Examples:

The formula:

$\exists x \ \exists y \ (x \ne y \land \forall z \ (z=x \lor z=y))$

is a sentence that has only finite models, i.e. it is satisfied in a domain with exactly two elemets.

Formally (see pages 81 and 83 for the notation):

$\vDash_{\mathfrak A } \exists x \ \exists y \ (x \ne y \land \forall z \ (z=x \lor z=y))$ iff $|\mathfrak A|$ has exactly two elements.

The formula:

$\forall x \lnot R(x,x) \land \forall x \forall y \forall z (R(x,y) \land R(y,z) \to R(x,z)) \land \forall x \exists y R(x,y)$

is satisfiable only in an infinite domain (interpret $R$ with $<$).