Nonfirstorderizable statements are sometimes presented as evidence that first-order logic is not adequate to capture the nuances of meaning in natural language.
[1] Quine argued that such sentences call for second-order symbolization, which can be interpreted as plural quantification over the same domain as first-order quantifiers use, without postulation of distinct "second-order objects" (properties, sets, etc.).
If Axy is understood to mean "x admires y," and the universe of discourse is the set of all critics, then a reasonable translation of the sentence into second order logic is:
states that there is a set X with the following three properties: Recall a model of a formal theory of arithmetic, such as first-order Peano arithmetic, is called standard if it only contains the familiar natural numbers as elements (i.e., 0, 1, 2, ...).
And then on the other hand, in every non-standard model there is a subset X satisfying the formula.
Let us now assume that there is a first-order rendering of the above formula called E. If
This is a contradiction, so we can conclude that no such formula E exists in first-order logic.
There is no formula A in first-order logic with equality which is true of all and only models with finite domains.
In other words, there is no first-order formula which can express "there is only a finite number of things".
[2] Suppose there is a formula A which is true in all and only models with finite domains.
We can express, for any positive integer n, the sentence "there are at least n elements in the domain".
For a given n, call the formula expressing that there are at least n elements Bn.
Applying the compactness theorem, the entire infinite set must also have a model.
This contradiction shows that there can be no formula A with the property we assumed.