Dedekind's help in his article. In previous articles, he had acknowledged help received from Kronecker, Weierstrass, Heine, and Hermann Schwarz. Cantor's failure to mention Dedekind's contributions damaged his relationship with Dedekind. Dedekind stopped replying to his letters and did not resume the correspondence until October 1876. The legacy of Cantor's article Cantor's article introduced the uncountability theorem and the concept of countability. Both would lead to significant developments in mathematics. The uncountability theorem demonstrated that one-to-one correspondences can be used to analyze infinite sets. In 1878, Cantor used them to define and compare cardinalities. He also constructed one-to-one correspondences to prove that the n-dimensional spaces Rn (where R is the set of real numbers) and the set of irrational numbers have the same cardinality as R. In 1883, Cantor extended the positive integers with his infinite ordinals. This extension was necessary for his work on the Cantor–Bendixson theorem. Cantor discovered other uses for the ordinals—for example, he used sets of ordinals to produce an infinity of sets having different infinite cardinalities. His work on infinite sets together with Dedekind's set-theoretical work created set theory. The concept of countability led to countable operations and objects that are used in various areas of mathematics. For example, in 1878, Cantor introduced countable unions of sets. In the 1890s, Émile Borel used countable unions in his theory of measure, and René Baire used countable ordinals to define his classes of functions. Building on the work of Borel and Baire, Henri Lebesgue created his theories of measure and integration, which were published from 1899 to 1901. Countable models are used in set theory. In 1922, Thoralf Skolem proved that if conventional axioms of set theory are consistent, then they have a countable model. Since this model is countable, its set of real numbers is countable. This consequence is called Skolem's paradox, and Skolem explained why it does not contradict Cantor's uncountability theorem: although there is a one-to-one correspondence between this set and the set of positive integers, no such one-to-one correspondence is a member of the model. Thus the model considers its set of real numbers to be uncountable, or more precisely, the first-order sentence that says the set of real numbers is uncountable is true within the model. In 1963, Paul Cohen used countable models to prove his independence theorems. See also Cantor's theorem Notes Note on Cantor's 1879 proof References Bibliography . .