Bernays and the completeness theorem

A well-known result in Reverse Mathematics is the equivalence of the formalized version of the Gödel completeness theorem [8] – i.e. every countable, consistent set of first-order sentences has a model – and Weak König's Lemma [WKL] – i.e. every infinite tree of 0-1 sequences contains an infinite path– over the base theory RCA0. It is less well known how the Completeness Theorem came to be studied in the setting of second-order arithmetic and computability theory. The first goal of this note will be to recount these developments against the backdrop of the latter phases of the Hilbert program, culminating in the publication of the second volume of Hilbert and Bernays’s [13] Grundlagen der Mathematiks in 1939. This work contains a detailed formalization of the Completeness Theorem in a system similar to first-order Peano arithmetic [PA] – a result which has come to be known as the Arithmetized Completeness Theorem. Its second goal will be to illustrate how reflection on this result informed Bernays’s views about the philosophy of mathematics, in particular in regard to his engagement with the maxim “consistency implies existence”