# [FOM] history of diagonal argument

Robert Black Mongre at gmx.de
Sun Jul 22 03:34:19 EDT 2007

```Does it have to be decimal? In the paper you
mention Cantor shows by diagonalization that
there are uncountably many formal binary
expansions. Then in §4 of the Beitraege (see pp.
288-289 of the Zermelo-edited Abhandlungen) he
argues that only countably many of the reals have
two binary expansions and that removing a
countable set from a set of power 2^Âleph_0
doesn't affect cardinality. See Zermelo's
editorial note on pp. 280-81.

Robert

>This is a question about the form of Cantor's diagonal argument as applied
>to decimal expansions of real numbers.
>The argument appears in
>
>@article{Cantor1890-91,
>author = "Georg Cantor",
>title = "{\"U}ber eine elementare {Frage} der {Mannigfaltigkeitslehre}",
>journal = "Jahresbericht der Deutschen Mathematischen Vereinigung",
>volume = "I",
>year = "1890--91",
>pages = "75--78"}
>
>It can be read on-line at
>
>Cantor's stated purpose is to exhibit uncountable infinities without
>using irrational numbers
>(unabh\"angig von der Betrachtung der Irrationalzahlen)
>and, indeed, no decimal expansions appear in the paper.
>
>Now I am perfectly willing to believe that Cantor must have realized
>that a proof of the uncountablity of the reals along these lines is
>possible but I have trawled through his collected works and I have not
>been able to find any explicit reference to a proof like this.
>I could also find no mention in Dauben's book on Cantor's mathematics.
>
>The earliest place I have been able to find the decimal-diagonalization
>proof is `Theory of Sets of Points' by Young and Young (1906),
>with a reference to the above paper.
>
>My question is twofold:
>- can someone point me to a place where Cantor explicitly wrote down
>   the decimal-diagonal argument or, again explicitly, mentioned the
>   possibility of such a proof?
>- failing that: is there a reference earlier than Young and Young's book
>   where this proof appears?
>
>Thanks,
>
>KP Hart
>
>--
>E-MAIL: K.P.Hart at TUDelft.NL           PAPER: Faculty EEMCS
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>                                              the Netherlands
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Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

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```