Burali-Forti Paradox

In the theory of transfinite ordinal numbers,

1. Every well ordered set has a unique ordinal number,

2. Every segment of ordinals (i.e., any set of ordinals arranged in natural order which contains all the predecessors of each of its elements) has an ordinal number which is greater than any ordinal in the segment, and

3. The set  of all ordinals in natural order is well ordered.

Then by statements (3) and (1),  has an ordinal . Since  is in , it follows that  by (2), which is a contradiction.

REFERENCES:

Burali-Forti, C. "Una questione sui numeri transfiniti." Rendiconti del Circolo Mat. di Palermo 11, 154-164, 1897.

Copi, I. M. "The Burali-Forti Paradox." Philos. Sci. 25, 281-286, 1958.

Curry, H. B. Foundations of Mathematical Logic. New York: Dover, p. 5, 1977.

Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 29-30, 1998.

Mirimanoff, D. "Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles." Enseign. math. 19, 37-52, 1917.