Georg Cantor and Transcendental Numbers

Kac and Ulam are comparing Cantor's method with Liouville's earlier construction of transcendentals [24, 25] they find Cantor's method to be non-constructive. But Herstein and Kaplansky insist that Cantor's method is constructive, that it can produce a transcendental. A few lines later, they assert that this transcendental "is as well determined a number as e or r." After reading the statements by Kac and Ulam, and Herstein and Kaplansky, we decided to study Cantor's work and how it has been presented. This article contains the results of our study. We begin by analyzing Cantor's original articles, his 1874 article that contains his first proof and his 1891 article that contains his diagonal proof. Our analysis will show that Cantor's methods lead to computer programs that generate transcendentals, and it will also determine which transcendentals are generated by the diagonal method. Next we will examine the history behind Cantor's first proof. Finally, we will consider how some commonly-held views about mathematics and its history have affected the interpretation of Cantor's work.