new language. Besides, the uninitiated may read it, thanks to an Italian interlinear translation.
What constitutes the importance of this memoir is that it has given the first example of those antinomies met in the study of transfinite numbers and making since some years the despair of mathematicians. The aim, says Burali-Forti, of this note is to show there may be two transfinite numbers (ordinals), a and b, such that a is neither equal to, greater than, nor less than b.
To reassure the reader, to comprehend the considerations which follow, he has no need of knowing what a transfinite ordinal number is.
Now, Cantor had precisely proved that between two transfinite numbers as between two finite, there can be no other relation than equality or inequality in one sense or the other. But it is not of the substance of this memoir that I wish to speak here; that would carry me much too far from my subject; I only wish to consider the form, and just to ask if this form makes it gain much in rigor and whether it thus compensates for the efforts it imposes upon the writer and the reader.
First we see Burali-Forti define the number 1 as follows:
a definition eminently fitted to give an idea of the number 1 to persons who had never heard speak of it.
I understand Peanian too ill to dare risk a critique, but still I fear this definition contains a petitio principii, considering that I see the figure 1 in the first member and Un in letters in the second.