Gödel's beta function, without the Chinese remainder theorem

See :

• Raymond Smullyan, Theory of formal systems (1961), page 78:

our proof follows novel lines in that all appeal to the traditional number theory devices accorded this in the past — e.g., prime factorization, congruences and the Chinese remainder theorem — are avoided. Thus Gödel's program of establishing incompleteness, even of first order theories involving plus and times as their sole arithmetical primitives, can, by the methods of this section, be carried out without appeal to number theory.