Published online by Cambridge University Press: 12 March 2014
Let C be an axiom system formalized within the first order functional calculus, and let C′ be related to C as the Bernays-Gödel set theory is related to the Zermelo-Fraenkel set theory. (An exact definition of C′ will be given later.) Ilse Novak [5] and Mostowski [8] have shown that, if C is consistent, then C′ is consistent. (The converse is obvious.) Mostowski has also proved the stronger result that any theorem of C′ which can be formalized in C is a theorem of C.
The proofs of Novak and Mostowski do not provide a direct method for obtaining a contradiction in C from a contradiction in C′. We could, of course, obtain such a contradiction by proving the theorems of C one by one; the above result assures us that we must eventually obtain a contradiction. A similar process is necessary to obtain the proof of a theorem in C from its proof in C′. The purpose of this paper is to give a new proof of these theorems which provides a direct method of obtaining the desired contradiction or proof.
The advantage of the proof may be stated more specifically by arithmetizing the syntax of C and C′.
The author wishes to thank the referee for various suggestions, especially concerning the advantages of the proof given here over previous proofs.