Journal of Symbolic Logic 19 (1):21-28 (1954)

LetCbe an axiom system formalized within the first order functional calculus, and letC′ be related toCas the Bernays-Gödel set theory is related to the Zermelo-Fraenkel set theory. Ilse Novak [5] and Mostowski [8] have shown that, ifCis consistent, thenC′ is consistent. Mostowski has also proved the stronger result that any theorem ofC′ which can be formalized inCis a theorem ofC.The proofs of Novak and Mostowski do not provide a direct method for obtaining a contradiction inCfrom a contradiction inC′. We could, of course, obtain such a contradiction by proving the theorems ofCone 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 inCfrom its proof inC′. 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 ofCandC′.
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DOI 10.2307/2267646
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References found in this work BETA

Mathematical Logic.Morton G. White & Willard Van Orman Quine - 1942 - Philosophical Review 51 (1):74.
Introduction to Mathematical Logic. Part I.E. N. - 1944 - Journal of Philosophy 41 (25):699.

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Incompleteness Via Paradox and Completeness.Walter Dean - 2020 - Review of Symbolic Logic 13 (3):541-592.
Predicative Logic and Formal Arithmetic.John P. Burgess & A. P. Hazen - 1998 - Notre Dame Journal of Formal Logic 39 (1):1-17.
Bernays and Set Theory.Akihiro Kanamori - 2009 - Bulletin of Symbolic Logic 15 (1):43-69.

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