Paradoxes of the Infinite
Transfinite theory is rife with paradoxes. As you wade through the volumes of literature on the subject, you are apt to find more written about the paradoxes than transfinite theory itself. The most famous paradox, of course, is the Russell Paradox.
The Russell Paradox is famous not only for being the first paradox associated to transfinite theory. It just happened to have been applied in a very dramatic style that resonates in the nightmares of all academicians.
For several years, the German mathematician Gottlob Frege worked on an epic volume (Grundgesetze der Arithmetik, 1902) that he hoped would serve as a consistent logical foundation for mathematics. Frege work was truly an ambitious project, and drew heavily on Cantor's transfinite theory for the definition of the Cardinal Numbers.
At the same time, the young English logician Bertrand Russell was interested in the liars paradox, and how it applied to meta mathematics. The young Russell heard of Frege's work. As Frege's volumes were going to to press, Russell wrote Frege and inquired how he resolved the reflexive paradox.
As it turns out. Frege had not given sufficient thought to the reflexive paradox, and Frege realized that the paradox opened a hole in several of his arguments.
The results were devastating. Despite the fact that Frege made a substantial number of important contributions in the development of predicate logic, and deserves credit as the father of logicism and mathematical logic. The myths loomed larger than fact. The image of Frege that history hands us is not of a great thinker who made substantial contributions to his field of study, but a haunted and pathetic creature who, on the day of his triumph, was cut down by a simple logical fallacy.
Rather than rehash the Russell Paradox, I thought I would render my own short version of the reflexive paradox in the essay titled "Namable Numbers."
B Carver at UCI has put together a rather comprehensive list Frege related resources at the site: http://www.ags.uci.edu/~bcarver/frege.html
©2002 Kevin Delaney