[FOM] history of diagonal argument

K. P. Hart K.P.Hart at tudelft.nl
Fri Jul 20 06:10:31 EDT 2007

This is a question about the form of Cantor's diagonal argument as applied
to decimal expansions of real numbers.
The argument appears in

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?


KP Hart

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