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.


"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 .

  • 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


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 $<$).


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