Cantor and continuity
Kanamori, Akihiro
Shapiro, Stewart
Georg Cantor (1845-1919), with his seminal work on sets and number, brought
forth a new field of inquiry, set theory, and ushered in a way of proceeding in
mathematics, one at base infinitary, topological, and combinatorial. While this
was the thrust, his work at the beginning was embedded in issues and concerns
of real analysis and contributed fundamentally to its 19th Century rigorization,
a development turning on limits and continuity. And a continuing engagement
with limits and continuity would be very much part of Cantor's mathematical
journey, even as dramatically new conceptualizations emerged. Evolutionary
accounts of Cantor's work mostly underscore his progressive ascent through settheoretic
constructs to transfinite number, this as the storied beginnings of set
theory. In this article, we consider Cantor's work with a steady focus on continuity,
putting it first into the context of rigorization and then pursuing the
increasingly set-theoretic constructs leading to its further elucidations.
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