Re: Is one-to-one mapping valid for comparing infinite-sized sets?

Horand.Gassm...@xxxxxxxxxxxxxx wrote:

This remark you need to explain. Since so many infinite
sets /can/ be put in bijection with N (for instance: Q,
the set of algebraic numbers, the set of polynomials with
rational coefficients, etc.) it was only natural in the
latter part of the nineteenth century to try and find a
bijection between N and R. What *was* surprising was the
*fact* that such as bijection is impossible.

This is not historically correct. The idea of there being
a bijection between the set of natural numbers and the set
of rational numbers was, with one exception that I'm aware of,
not considered by anyone except Cantor (in late November 1873).
In letters to Dedekind, Cantor mentioned this result, saying
that it is not difficult to prove once one's mind brings it
under consideration (or something to this effect), and
invited Dedekind to find a proof, which Dedekind did.
In fact, Dedekind proved the stronger result that the set
of natural numbers can be put into one-to-one correspondence
with the set of algebraic numbers, and to show this stronger
result Dedekind made use of the idea of the height of a polynomial.
Cantor wound up putting Dedekind's theorem and proof in Cantor's
1874 paper, which irked Dedekind a little since Cantor didn't
mention that Dedekind suggested both the result and the method
of proof (but this was a minor rift in their long friendship
and correspondence). In his letters to Dedekind during Dec. 1873,
Cantor said he assumed the same would be true for the real numbers,
but Cantor was unable to come up with a proof and he asked if
Dedekind had any ideas of how to prove or disprove this (that
the natural numbers can be put in one-to-one correspondence with
the real numbers), and Dedekind more than once said he tried
proving or disproving the assertation without success. Then
Cantor announced in a letter to Dedekind (also in Dec. 1873,
but I don't know the exact day right now) that he (Cantor) had
found a way to disprove such a one-to-one correspondence.
[No, this 1873-74 proof wasn't Cantor's diagonalization proof.
That was presented in at a Sept. 1891 conference and published
in 1892.]

The only reference I've come across that anyone before
Cantor had considered the idea of trying to establish
a one-to-one correspondence between the set of natural numbers
and the set of rational numbers are some comments on p. 99
(footnote 1) of the paper listed below. Apparently, Karl
Weierstrass brought up the idea in an 1873-74 seminar (I think
it's not known if Weierstrass was in any way influenced by
prior contact with Cantor, but I'm not sure about this) and
I believe one student managed to show the result. ["I think"
and "I believe", because I'm going on memory of what the passage
on p. 99 says, not being able to read German. I've had it
translated and incorporated into some notes of mine, but my
notes are not presently available to me.]

Arthur Schoenflies, "Zur Erinnerung an Georg Cantor" [To the
memory of Georg Cantor], Jahresberichte der Deutschen
Mathematiker-Vereinigung 31 (1922), 97-106.
http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=243972
http://gdz.sub.uni-goettingen.de/no_cache/dms/load/toc/?IDDOC=248948

Dave L. Renfro
.

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