Countable Nonstandard Models: Following Skolem’s Approach

In 1934, Skolem gave a remarkable construction of a countable nonstandard model of arithmetic. His construction contains ideas of the ultrapower construction which was introduced in model theory 20 years later. However, typical ultrapower constructions produce uncountable models. Skolem’s construction can also be connected with ideas from computability theory, formalized by Turing and others in 1936. The proof of one of Skolem’s key statements can be interpreted using computability-theoretic notions of indecomposable sets developed in the late 1950s. We favor the computability-theoretic approach and describe the connections of Skolem’s ideas with our current study of cohesive powers of models in computable model theory.