“What fermented in me for years”: Cantor's discovery of transfinite numbers

✍ Scribed by José Ferreirós

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 638 KB
• Volume: 22
• Category: Article
• ISSN: 0315-0860
• DOI: 10.1006/hmat.1995.1003

✦ Synopsis

Transfinite (ordinal) numbers were a crucial step in the development of Cantor's set theory. The new concept depended in various ways on previous problems and results, and especially on two questions that were at the center of Cantor's attention in September 1882, when he was about to make his discovery. First, the proof of the Cantor-Bendixson theorem motivated the introduction of transfinite numbers, and at the same time suggested the "principle of limitation," which is the key to the connection between transfinite numbers and infinite powers. Second, Dedekind's ideas, which Cantor discussed in September 1882, seem to have played an important heuristic role in his decision to consider the "symbols of infinity" that he had been using as true numbers, i.e., as autonomous objects; to this end Cantor introduced in his work, for the first time, ideas on (well) ordered sets.