Editing Elementary class (section)

=== An elementary, basic pseudoelementary class that is not basic elementary ===
Let σ be an arbitrary signature. The class ''K'' of all infinite σ-structures is elementary. To see this, consider the sentences

:<math>\rho_2={}</math> "<math>\exist x_1\exist x_2(x_1 \not =x_2)</math>",

:<math>\rho_3={}</math> "<math>\exist x_1\exist x_2\exist x_3((x_1 \not =x_2) \land (x_1 \not =x_3) \land (x_2 \not =x_3))</math>",

and so on. (So the sentence <math>\rho_n</math> says that there are at least ''n'' elements.) The infinite σ-structures are precisely the models of the theory

:<math>T_\infty=\{\rho_2, \rho_3, \rho_4, \dots\}</math>.

But ''K'' is not a basic elementary class. Otherwise the infinite σ-structures would be precisely those that satisfy a certain first-order sentence τ. But then the set
<math>\{\neg\tau, \rho_2, \rho_3, \rho_4, \dots\}</math> would be inconsistent. By the [[compactness theorem]], for some natural number ''n'' the set <math>\{\neg\tau, \rho_2, \rho_3, \rho_4, \dots, \rho_n\}</math> would be inconsistent. But this is absurd, because this theory is satisfied by any finite σ-structure with <math>n+1</math> or more elements.

However, there is a basic elementary class ''K<nowiki>'</nowiki>'' in the signature σ' = σ <math>\cup</math> {''f''}, where ''f'' is a unary function symbol, such that ''K'' consists exactly of the reducts to σ of σ'-structures in ''K<nowiki>'</nowiki>''. ''K<nowiki>'</nowiki>'' is axiomatised by the single sentence <math>(\forall x\forall y(f(x) = f(y) \rightarrow x=y) \land \exists y\neg\exists x(y = f(x))),</math>, which expresses that ''f'' is injective but not surjective. Therefore, ''K'' is elementary and what could be called basic pseudo-elementary, but not basic elementary.