Some new properties of transfinite ordinals

1 Presented to the Society, December 27, 1939. The author is indebted to Professor W. A. Hurwitz for his indispensable advice in the preparation of this paper. 2 W. Sierpinski, Leçons sur les Nombres Transfinis, Paris, 1928, p. 202; hereafter referred to as Sierpinski (I). The theorem was first stated for ordinals of the first and second ordinal class by G. Cantor, Beitrdge zur Begründung der transfiniten Mengenlehre, Mathematische Annalen, vol. 49 (1897), p. 237, and for all ordinals by G. Hessenberg, Grundbegriffe der Mengenlehre, Abhandlungen der Fries'schen Schule, New Series 1.4, Göttingen, 1906, p. 587. 3 W. Sierpinski, A property of ordinal numbers, Bulletin of the Calcutta Mathematical Society, vol. 20 (1930), pp. 21-22; hereafter referred to as Sierpinski (II). 4 A set of ordinal numbers is said to be closed if it contains the least upper bound of each subset. 5 F . Siecza, Sur Vunicité de la décomposition de nombres ordinaux en facteurs irréductibles, Fundamenta Mathematicae, vol. 5 (1924), pp. 172-175.