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Julius Wilhelm Richard Dedekind (1831 - 1916) redefined the notion of irrational numbers
in terms of Dedekind cuts, and thus gave a construction of the real numbers. He was a good
friend of Cantor, and encouraged him to publish, even though the results seemed counter-intuitive.
He also introduced the notion of an ideal which is fundamental to ring theory.
One of the Famous People on this site.