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==The article==
Cantor's article is short, less than four and a half pages.{{efn-ua|In letter to Dedekind dated December 25, 1873, Cantor states that he has written and submitted "a short paper" titled ''On a Property of the Set of All Real Algebraic Numbers''.
({{harvnb|Noether|Cavaillès|1937|p=17}}; English translation: {{harvnb|Ewald|1996|p=847}}.)}} It begins with a discussion of the real [[algebraic number]]s and a statement of his first theorem: The set of real algebraic numbers can be put into [[one-to-one correspondence]] with the set of positive integers.<ref name=Cantor1874>{{harvnb|Cantor|1874}}. English translation: {{harvnb|Ewald|1996|pp=840–843}}.</ref> Cantor restates this theorem in terms more familiar to mathematicians of his time: "The set of real algebraic numbers can be written as an infinite [[sequence]] in which each number appears only once."<ref name=Gray828>{{harvnb|Gray|1994|p=828}}.</ref>
Cantor's second theorem works with a [[closed interval]] [''a'', ''b''], which is the set of real numbers ≥ ''a'' and ≤ ''b''. The theorem states: Given any sequence of real numbers ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ... and any interval [''a'', ''b''], there is a number in [''a'', ''b''] that is not contained in the given sequence. Hence, there are infinitely many such numbers.<ref name=Ewald840_841>{{harvnb|Cantor|1874|p=259}}. English translation: {{harvnb|Ewald|1996|pp=840–841}}.</ref>
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