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The Burali-Forti paradox

Published online by Cambridge University Press:  12 March 2014

Barkley Rosser*
Cornell University


In the system presented by Quine in his book Mathematical logic, one can derive the Burali-Forti Paradox. It is the purpose of this paper to present the details of this derivation. For the derivation, only familiarity with Quine's book is assumed.

The present derivation is based on the derivations given by Hobson and by Whitehead and Russell.

In both these sources, the primary interest is the theory of ordinals, and the Burali-Forti Paradox is of interest only as something to be avoided. In the present paper the primary interest is the Burali-Forti Paradox, so that much of the theory of ordinals is absent from this paper and only those details remain which are relevant to the derivation of the paradox.

Research Article
Copyright © Association for Symbolic Logic 1942

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1 W. V., Quine, Mathematical logic, New York 1940.Google Scholar

2 It has come to my attention that there is a question of priority connected with the discovery that Quine's system admits the Burali-Forti Paradox. Quine tells me that a former student of his, Mr. Roger C. Lyndon, while studying the theory of ordinals in Quine's system, came upon the Burali-Forti Paradox. This happened in the latter half of October. Mr. Lyndon's first reaction to his discovery was that it must be the result of an error on his part. After an unsuccessful effort to find such an error, Mr. Lyndon sent his proof to Quine in December. Quine describes the proof as painstaking, detailed, and correct. Hence Mr. Lyndon certainly deserves credit for independent discovery of the paradox.

In my case, the circumstances are as follows. Toward the end of September, I wrote Quine to the effect that I had been unable to convince myself that his system did not admit the Burali-Forti Paradox, and suggested that he look into the matter. Somewhat later, Quine wrote back that he was busy, and requested me to investigate carefully. Still later, I sent Quine an earlier draft of the present paper. I happen to have kept his reply, which was fairly prompt, and was dated October 24, and stated that my manuscript undoubtedly established the presence of the paradox.

Obviously, we must fix the date of my discovery of the paradox as being the date on which I prepared that manuscript, rather than the earlier date on which I suspected the presence of the paradox.

On the basis of the above data, it would seem equitable to say that my discovery of the paradox and Mr. Lyndon's discovery were simultaneous.

3 Hobson, E. W., The theory of functions of a real variable, first edition, Cambridge, England, 1907Google Scholar, see Chapter III; second edition, 1921, and third edition, 1927, see Chapter IV of the first volume.

4 Whitehead, A. N. and Russell, Bertrand, Principia mathematica, volumes 2 and 3, see *150*152Google Scholar, *154, *155, *160, *161, *180, *181, *200–*202, *204–*208, *210–*214, *250–*256.

5 Quine, W. V., New foundations for mathematical logic, The American mathematical monthly, vol. 44 (1937), pp. 7080.CrossRefGoogle Scholar We shall refer to this paper as Quine I.

6 Quine, W. V., On the theory of types, this JOURNAL, vol. 3 (1938), pp. 125139.Google Scholar We shall refer to this paper as Quine II.

7 This paper has since appeared, in this JOURNAL, vol. 6 (1941), pp. 135–149. Editor'.