Incompleteness and Undecidability
Peter G. Hinman · 2018
In Sections 2.4-2.5 we worked out solutions to several instances of the problem: given a structure A, find a set Γ of sentences in the language of A which axiomatize the theory Th(A) of A — that is, such that Th(A) = Th(Γ). This has been considered a natural question throughout the history of mathematics because it provides a deductive structure for a theory of mathematical interest. Of course, before the development of the techniques of logic, the problem could not be stated in this sharp form, but we may see even Euclid’s axiomatization of geometry as a first example.