There is no first-order sentence which is true in all and only finite models and not in any infinite models.
Sketch of conventional proof: The compactness theorem says that if a collection of first-order sentences is inconsistent, then a finite subset of those first-order sentences is inconsistent.
To a sentence or theory true of all finite sets, adjoin the infinite series of statements “This model has at least one element”, “This model has at least two elements” (that is, there exist a and b with a != b), “This model has at least three elements” (the finite sentence: exists a, b, c, and a != b, b != c, a != c), and so on.
No finite subset of these statements is inconsistent with the original theory, therefore by compactness the set as a whole is consistent with the original theory. Therefore the original theory possesses an infinite model. QED.
There is no first-order sentence which is true in all and only finite models and not in any infinite models.
Sketch of conventional proof: The compactness theorem says that if a collection of first-order sentences is inconsistent, then a finite subset of those first-order sentences is inconsistent.
To a sentence or theory true of all finite sets, adjoin the infinite series of statements “This model has at least one element”, “This model has at least two elements” (that is, there exist a and b with a != b), “This model has at least three elements” (the finite sentence: exists a, b, c, and a != b, b != c, a != c), and so on.
No finite subset of these statements is inconsistent with the original theory, therefore by compactness the set as a whole is consistent with the original theory. Therefore the original theory possesses an infinite model. QED.