Page "Georg Cantor" Paragraph 25

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 article, " Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen " (" On a Property of the Collection of All Real Algebraic Numbers ").

This article was the first to provide a rigorous proof that there was more than one kind of infinity.

Previously, all infinite collections had been implicitly assumed to be equinumerous ( that is, of " the same size " or having the same number of elements ).

Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous.

In other words, the real numbers are not countable.

His proof is more complex than the more elegant diagonal argument that he gave in 1891.

Cantor's article also contains a new method of constructing transcendental numbers.

Transcendental numbers were first constructed by Joseph Liouville in 1844.