A formula with no recursively enumerable model

Publisher Summary This chapter gives example of a formula that has no recursive model. This example seems to be simpler as it makes no reference to the axiomatic set-theory and uses exclusively tools known from the theory of recursive functions. The formula given in the chapter was found in the course of unsuccessful attempts to construct a formula, no model of which would belong to the smallest field of sets generated by the various classes. It has been observed that many recent papers in the field of symbolic logic do not supply full proofs of the statements they contain. While it would certainly not be reasonable to require from all papers to give exhaustive proofs, it is certainly necessary to publish full proofs from time to time.