1
$\begingroup$

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 .

$\endgroup$
  • 1
    $\begingroup$ Do you know what a model is? "Finite" and "infinite" refer to the size of the set of individuals in the model. $\endgroup$ – Henning Makholm Apr 11 '17 at 11:23
4
$\begingroup$

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 $<$).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.