Abstract Set Theory

[from the Introduction]

"I protest… against the use of infinite magnitude as if it were something finished; this use is not admissible in mathematics. The infinite is only a façon de parler: one has in mind limits approached by certain ratios as closely as desirable while other ratios may increase indefinitely. C.F. Gauss, presumably the foremost mathematician of the 19th century, expressed this view in 1831 in reply to an idea of Schumacher's and hereby uttered a horror infiniti which up to almost the end of the century was the common attitude of mathematicians and seemed unassailable considering the authority of Gauss. Mathematics should deal with finite magnitudes and finite numbers only while the treatment of actual infinity, whether infinitely great or small, might be left to philosophy.

It was the mathematician Georg Cantor (1845-1918) who dared to fight this attitude and, in the opinion of the majority of 20th century mathematicians, has succeeded in the task of bestowing legitimacy upon infinitely great magnitude. …