Editing Georg Cantor (section)

===Number theory, trigonometric series and ordinals===
Cantor's first ten papers were on [[number theory]], his thesis topic. At the suggestion of [[Eduard Heine]], the Professor at Halle, Cantor turned to [[Mathematical analysis|analysis]]. Heine proposed that Cantor solve [[Open problem|an open problem]] that had eluded [[Peter Gustav Lejeune Dirichlet]], [[Rudolf Lipschitz]], [[Bernhard Riemann]], and Heine himself: the uniqueness of the representation of a [[Function (mathematics)|function]] by [[trigonometric series]]. Cantor solved this problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices ''n'' in the ''n''th [[derived set (mathematics)|derived set]] ''S''<sub>''n''</sub> of a set ''S'' of zeros of a trigonometric series. Given a trigonometric series f(x) with ''S'' as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had ''S''<sub>1</sub> as its set of zeros, where ''S''<sub>1</sub> is the set of [[limit point]]s of ''S''. If ''S''<sub>''k+1''</sub> is the set of limit points of ''S''<sub>''k''</sub>, then he could construct a trigonometric series whose zeros are ''S''<sub>''k+1''</sub>. Because the sets ''S''<sub>''k''</sub> were closed, they contained their limit points, and the intersection of the infinite decreasing sequence of sets ''S'', ''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>,... formed a limit set, which we would now call ''S''<sub>''ω''</sub>, and then he noticed that ''S''<sub>ω</sub> would also have to have a set of limit points ''S''<sub>ω+1</sub>, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers ''ω'', ''ω''&nbsp;+&nbsp;1, ''ω''&nbsp;+&nbsp;2, ...<ref>{{Cite journal |last1=Cooke|first1=Roger|title=Uniqueness of trigonometric series and descriptive set theory, 1870–1985|journal=Archive for History of Exact Sciences| volume=45|page=281|year=1993|doi=10.1007/BF01886630|issue=4|s2cid=122744778}}</ref>

Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining [[irrational number]]s as [[Sequence space|convergent sequences]] of [[rational number]]s. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by [[Dedekind cut]]s.  While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of [[infinitesimal]]s of his contemporaries [[Otto Stolz]] and [[Paul du Bois-Reymond]], describing them as both "an abomination" and "a [[cholera]] [[bacillus]] of mathematics".<ref name="Infinitesimal">{{Cite journal |author=Katz, Karin Usadi |author2=Katz, Mikhail G. |author2-link=Mikhail Katz |year=2012|title= A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography|journal= [[Foundations of Science]]|doi=10.1007/s10699-011-9223-1|volume =17|number=1|pages=51–89|arxiv=1104.0375|s2cid=119250310 }}</ref> Cantor also published an erroneous "proof" of the inconsistency of [[infinitesimal]]s.<ref>{{Cite journal |author=Ehrlich, P.| year=2006| title=The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes|journal=Arch. Hist. Exact Sci.| volume=60|number=1|pages=1–121| url=http://www.ohio.edu/people/ehrlich/AHES.pdf|doi=10.1007/s00407-005-0102-4| s2cid=123157068|url-status=dead| archive-url=https://web.archive.org/web/20130215061415/http://www.ohio.edu/people/ehrlich/AHES.pdf|archive-date=15 February 2013}}</ref>