Ultrasets and the Paradoxes of Set Theory

Abstract

In nontechnical German, rather paradoxically, sets and numbers (Mengen and Zahlen) that are exceedingly large are called Unmengen and Unzahlen — literally, non-sets and non-numbers. The closest English analogue that occurs to me is the word invaluable for something that is exceedingly valuable. In what follows, I translate Unmengen and Unzahlen by “ultrasets” and “ultranumbers”. Actually I did not take this little note quite seriously in 1927 when writing it. It appeared as the third in the series [1928, 1]. I here include a (condensed) translation of the note because it demonstrates in an utterly elementary fashion the need for the distinction between two kinds of totalities (here called sets and ultrasets) now generally made in the axiomatics of set theory between sets and classes.