The celebrated theorem of Löwenheim and Skolem tells us that every consistent set S of quantificational schemata (i.e., every set of well-formed formulas of the lower predicate calculus admitting of a true interpretation in some non-empty universe) admits of a true numerical interpretation (i.e., an interpretation of predicate letters such that all schemata of S come out true when the variables of quantification are construed as ranging over just the positive integers).

Later literature goes farther, and shows how, given S, actually to produce a numerical interpretation which will fit S in case S is consistent. The general case is covered by Kleene (see Bibliography). The special case where S contains just one schema (or any finite number, since we can form their conjunction) had been dealt with by Hilbert and Bernays. Certain extensions, along lines not to be embarked on here, have been made by Kleene, Kreisel, Hasenjäger, and Wang.

My present purpose is expository: to make the construction of the numerical interpretation, and the proof of its adequacy, more easily intelligible than they hitherto have been. The reasoning is mainly Kleene's, though closer in some ways to earlier reasoning of Gödel.