# [FOM] history of diagonal argument

K. P. Hart K.P.Hart at tudelft.nl
Mon Jul 23 12:27:26 EDT 2007

```You and Zermelo make a point that I already conceded, namely that the
diagonal argument is easily converted to a proof that R is uncountable.
However, in section 4 of the Beitr\"age Cantor is quite excited about the
fact that he can prove that R and R^2 have the same cardinality `mit diesen
wenigen Strichen' yet he says nothing about thus finding a new way of proving
the reals uncountable.
And that is exactly what I am looking for: a place where Cantor himself wrote
down that the diagonal argument can be used to prove that R is uncountable,
be it using binary or decimal digits.

KP

On Sun, Jul 22, 2007 at 09:34:19AM +0200, Robert Black wrote:
> 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^?leph_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
> >PHONE:  +31-15-2784572                       TU Delft
> >FAX:    +31-15-2787245                       Postbus 5031
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> >                                              the Netherlands
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>
> --
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>
> Robert Black
> Dept of Philosophy
> University of Nottingham
> Nottingham NG7 2RD
>
> tel. 0115-951 5845
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--
E-MAIL: K.P.Hart at TUDelft.NL           PAPER: Faculty EEMCS
PHONE:  +31-15-2784572                       TU Delft
FAX:    +31-15-2787245                       Postbus 5031
URL:    http://fa.its.tudelft.nl/~hart       2600 GA  Delft
the Netherlands
```