A theorem on the extendability of certain subsets of a Boolean algebra to ultrafilters which preserve countably many infinite meets (generalizing Rasiowa-Sikorsld) is used to pinpoint the mechanism of the Barwise proof in a way which bypasses the set theoretical elaborations.
Technical summary
By replacing each universal, resp. existential, quantification with the infinite conjunction, resp. disjunction, over all instances of a formula resulting fiom substituting for the quantified variable every other variable, one converts every formula in ~:r into an open formula, i.e., one in ~o. This conversion commutes with the propositional connectives, i.e., furnishes a propositional interpretation of/:~ in Z:~; moreover, the same surjective assignments of elements to variables will validate the converted formula as did the original one. Thus, in considering models of cardinality not exceeding the number of variables, one can replace the closed formulae of/~oo~ by their open conversions in /:~; and beyond that, by their classes modulo the "semantic" equivalence of being satisfied by the same surjective assignments to variables of elements in all appropriate relational systems. Now the classes of this equivalence constitute a complete Boolean algebra (under the induced action of the propositional connectives); any subset of formulae, dosed under the fmitary connectives, maps on a (partially in) fmitary Boolean subalgebra; and if this subset includes all the atomic formulae then the relational structures of caxdinality not exceeding that of the variables and having interpretations also for the infinitaxy propositional combinations in the subset, correspond to ultrafilters preserving the images of these combinations in the image Boolean subalgebra. This permits the application of Boolean techniques to the construction of models for such 'Tragments" of L:oo~. (Cf. Keisler, Makkai.) 1During the preparation of this work the second author was a postdoctoral visitor at the C. R. M.