On Cantor's Diagonalization Argument
Abstract
Although mathematicians had for some time expected that
differences observed between the natural and real numbers
were in part attributable to the relative size of the sets
in question, it was not until 1874 that German mathematician
Georg Cantor published a proof, tucked away in a misleadingly
titled article, of the now famous result that the reals are
uncountable. This first argument establishing the
nondenumerability of the continuum, however, was far from the
last, and in 1891 a dissatisfied Cantor produced a more elegant
proof in which he premiered the diagonalization method today
familiar to every student of real analysis. And while the result
itself continues to be a classic-new proofs of this foundational
fact about the cardinality of sets appear with some frequency-Cantor's
diagonalization procedure has proven a useful tool for everything
from constructing tight mathematical arguments to understanding
the self-referential paradoxes that have perplexed mankind for millennia.
Table of Contents
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The Bulletin of Symbolic Logic, Vol. 9, No. 3 (Sept. 2003), 362—386.
