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==History==
Cantor gave essentially this proof in a paper published in 1891 "Über eine elementare Frage der Mannigfaltigkeitslehre",<ref>{{Citation|language=de|first=Georg|last=Cantor|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002113910&physid=PHYS_0084 <!--http://resolver.sub.uni-goettingen.de/purl?GDZPPN002113910-->|title=Über eine elementare Frage der Mannigfaltigskeitslehre|journal=Jahresbericht der Deutschen Mathematiker-Vereinigung|volume=1|year=1891|pages=75–78}}, also in ''Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts'', E. Zermelo, 1932.</ref> where the [[Cantor's diagonal argument|diagonal argument]] for the uncountability of the [[real number|reals]] also first appears (he had [[Cantor's first uncountability proof|earlier proved the uncountability of the reals by other methods]]). The version of this argument he gave in that paper was phrased in terms of indicator functions on a set rather than subsets of a set.<ref>A. Kanamori, "[https://math.bu.edu/people/aki/8.pdf The Empty Set, the Singleton, and the Ordered Pair]", p.276. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.</ref> He showed that if ''f'' is a function defined on ''X'' whose values are 2-valued functions on ''X'', then the 2-valued function ''G''(''x'') = 1 − ''f''(''x'')(''x'') is not in the range of ''f''.
[[Bertrand Russell]] has a very similar proof in ''[[Principles of Mathematics]]'' (1903, section 348), where he shows that there are more [[propositional function]]s than objects. "For suppose a correlation of all objects and some propositional functions to have been affected, and let phi-''x'' be the correlate of ''x''. Then "not-phi-''x''(''x'')," i.e. "phi-''x'' does not hold of ''x''" is a propositional function not contained in this correlation; for it is true or false of ''x'' according as phi-''x'' is false or true of ''x'', and therefore it differs from phi-''x'' for every value of ''x''." He attributes the idea behind the proof to Cantor.
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