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Almost at the same time that Weierstrass was successfully banishing
infinitesimals and the infinite from definitions of limits (these two
notions had been the main tools for discussing limits in the seventeenth
and eighteenth centuries), Cantor and Dedekind were in correspondence
about infinitests and thepossibility of transfinitecadinals. Cantor's
first paper on the subject was published in 1874 and contained a proof
that the set of algebraic numbers (solutions of polynomial equations
with integer coeMcients) was countably infinite while the set of real
numbers was not. Modestly, he claimed that this provided a new proof
that transcendental numbers were dense, which had been shown by
Liouville in 1851. The argument of qn 23 was given by Cantor in 1891.
In 1879, Cantor introduced the notion of a dense set of numbers. It was
in 1885 that A. Harnack showed that a countable infinity of points did
not occupy length on the line.
