Editing Nonfirstorderizability (section)

=== Geach-Kaplan sentence ===
A standard example is the ''[[Peter Geach|Geach]]–[[David Kaplan (philosopher)|Kaplan]] sentence'': "Some critics admire only one another."
If ''Axy'' is understood to mean "''x'' admires ''y''," and the [[universe of discourse]] is the set of all critics, then a reasonable [[logic translation|translation of the sentence]] into second order logic is:
<math display="block">\exists X \big( (\exists x \neg Xx) \land \exists x,y (Xx \land Xy \land Axy) \land \forall x\, \forall y (Xx \land Axy \rightarrow Xy)\big)</math>
In words, this states that there exists a collection of critics with the following properties: The collection forms a proper subclass of all the critics; it is inhabited (and thus non-empty) by a member that admires a critic that is also a member; and it is such that if any of its members admires anyone, then the latter is necessarily also a member.

That this formula has no first-order equivalent can be seen by turning it into a formula in the language of arithmetic. To this end, substitute the formula <math display="inline"> ( y = x + 1 \lor x = y + 1 ) </math> for ''Axy''. This expresses that the two terms are successors of one another, in some way. The resulting proposition,
<math display="block">\exists X \big( (\exists x \neg Xx) \land \exists x,y (Xx \land Xy \land (y = x + 1 \lor x = y + 1)) \land \forall x\, \forall y (Xx \land (y = x + 1 \lor x = y + 1) \rightarrow Xy)\big)</math>
states that there is a set {{mvar|X}} with the following three properties:
* There is a number that does not belong to {{mvar|X}}, i.e. {{mvar|X}} does ''not contain all'' numbers.
* The set {{mvar|X}} is inhabited, and here this indeed immediately means there are at least two numbers in it.
* If a number {{mvar|x}} belongs to {{mvar|X}} and if {{mvar|y}} is either {{math|x + 1}} or {{math|x - 1}}, then {{mvar|y}} also belongs to {{mvar|X}}.
Recall a model of a formal theory of arithmetic, such as [[Peano axioms#Peano arithmetic as first-order theory|first-order Peano arithmetic]], is called ''standard'' if it ''only'' contains the familiar natural numbers as elements (i.e., {{math|0, 1, 2, ...}}). The model is called [[Non-standard model of arithmetic|non-standard]] otherwise. The formula above is true only in non-standard models: In the standard model {{mvar|X}} would be a proper subset of all numbers that also would have to contain all available numbers ({{math|0, 1, 2, ...}}), and so it fails. And then on the other hand, in every non-standard model there is a subset {{mvar|X}} satisfying the formula.

Let us now assume that there is a first-order rendering of the above formula called {{mvar|E}}. If <math>\neg E</math> were added to the Peano axioms, it would mean that there were no non-standard models of the augmented axioms. However, the usual argument for the [[Non-standard model of arithmetic#From the compactness theorem|existence of non-standard models]] would still go through, proving that there are non-standard models after all. This is a contradiction, so we can conclude that no such formula {{mvar|E}} exists in first-order logic.