The infinite-model categoricity test for completeness

If $T$ were not complete in the sense of F3, some sentence $\sigma$ would satisfy $T⊬\sigma$ and $T⊬\mathrm{\neg }\sigma$. By F4 these mean $T\models ̸\sigma$ and $T\models ̸\mathrm{\neg }\sigma$. Thus there are models $A,B$ of $T$ with $A\models \mathrm{\neg }\sigma$ and $B\models \sigma$, respectively. The hypothesis on $T$ makes both infinite.

For each of $A,B$, if its size is at most $\kappa$, apply F1; if it is at least $\kappa$, apply F2 with empty parameter set. Since the language is countable and $\kappa$ infinite, both language bounds hold. Under A1 this produces $A^{\mathrm{\prime }},B^{\mathrm{\prime }}$ of size exactly $\kappa$, elementarily equivalent respectively to $A,B$. At equality take the structure itself. Consequently $A^{\mathrm{\prime }}\models \mathrm{\neg }\sigma$ and $B^{\mathrm{\prime }}\models \sigma$, and both satisfy $T$.

Categoricity gives an isomorphism $h:A^{\mathrm{\prime }}\to B^{\mathrm{\prime }}$. It preserves values of terms by induction on terms: variables and constants are preserved, and each function commutes with $h$. Hence it preserves and reflects equality and relation atoms. Negation and conjunction retain this equivalence; existential witnesses transfer forward by $h$ and backward by its inverse. Formula induction therefore makes isomorphic structures agree on every sentence, contradicting their opposite decisions of $\sigma$. No such undecided sentence exists, so $T$ is complete.