diagonal argument

[K. P. Hart]

Contrary to popular belief Cantor's (original) diagonal argument did not
involve decimal
expansions of real numbers, see, e.g.,
http://uk.geocities.com/fr...@btinternet.com/cantor/diagarg.htm
Also his stated purpose was not to demonstrate the uncountability of the
real line
but to show that there are larger magnitudes than the countable,
*without* involving
irrational numbers.

What I am interested in is learning who actually first diagonalized
decimal expansions.
Any pointers are appreciated.

K. P. Hart

[Dave Seaman]

According to a previous thread in sci.math, Cantor was the first to
describe what is now called the Cantor set in terms of base-3
representations. This establishes a natural bijection between a subset
of the real line and the power set of the naturals. I don't know the
exact history, but it seems likely that this observation may have helped
lead to the second diagonal proof, the one involving decimal digits.

--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>

[K. P. Hart]

Dave Seaman wrote:

>On Tue, 13 Dec 2005 16:00:00 +0100, K. P. Hart wrote:
>
>
>>Contrary to popular belief Cantor's (original) diagonal argument did not
>>involve decimal
>>expansions of real numbers, see, e.g.,
>>http://uk.geocities.com/fr...@btinternet.com/cantor/diagarg.htm
>>Also his stated purpose was not to demonstrate the uncountability of the
>>real line
>>but to show that there are larger magnitudes than the countable,
>>*without* involving
>>irrational numbers.
>>
>>
>
>
>
>>What I am interested in is learning who actually first diagonalized
>>decimal expansions.
>>Any pointers are appreciated.
>>
>>
>
>According to a previous thread in sci.math, Cantor was the first to
>describe what is now called the Cantor set in terms of base-3
>representations. This establishes a natural bijection between a subset
>of the real line and the power set of the naturals. I don't know the
>exact history, but it seems likely that this observation may have helped
>lead to the second diagonal proof, the one involving decimal digits.
>
>
>

That's my point: he did *not* use decimals in the diagonal argument;
follow the link that I supplied: ``we take two symbols m and w and consider
the set of all sequences of these symbols ...''
I want to know who converted Cantor's `m' and `w' into decimals in order
to bring the proof back to the reals, so to speak.

KP

[abe.buc...@gmail.com]

It's a pretty natural extention, probably every math professor trying
to excite first year students did it independently.

[Dave Seaman]

No, *you* need to read what *I* wrote. I didn't say he used decimals; I
said he used *base-3* representations in proving that the Cantor set is
uncountable. A real number x in [0,1] belongs to the Cantor set iff it
has a base-3 representation that uses only the digits 0 and 2.

There is an obvious bijection between the set of all characteristic
functions on the naturals and the set of all mappings f: N -> {0,2}.

Moving from base 3 to base 10 does not seem like such a huge leap to me.

[Achava Nakhash, the Loving Snake]

A most interesting article. Cantor maintains that the uncountability
of the real numbers in any finite interval follows from this proof, and
of course it does, so the concept of using decimal or any other
expansions must have already occurred to him. Indeed the reals whose
decimal expansions contain only 2 and 5, or 4 and 7, or whatever, are
shown to be uncountable and the reals along with them, by this
argument. This approach also avoids all of the silliness involving
.99999... = 1 that must be accounted for in listing all of the reals by
decimal expansion. Thus it seems to me that Cantor effectively did it
in this article but presented it in a more general way.

Regards,
Achava

[Dave L. Renfro]

K. P. Hart wrote:

I'm pretty sure the decimal diagonal argument is in
Borel's 1898 text "Leçons sur la Théorie des Fonctions",
but my (photo) copy of it is at home so I can't check on
this right now. However, the decimal argument does *not*
appear in Borel's 1895 paper "Sur quelques points de la
théorie des fonctions", on the internet at

http://www.numdam.org/item?id=ASENS_1895_3_12__9_0
http://www.emis.de/cgi-bin/JFM-item?26.0429.03

which was an earlier version of part 2 of Borel's 1898
book. This 1895 paper was the published version of
Borel's 1894 Ph.D. Dissertation, by the way.

You might also want to look at two surveys Cantor
published in the mid 1890's. The text of each is on
the internet, but the papers are in German (which I
can't read) and I don't have time to look at them
right now in any event.

http://www.emis.de/cgi-bin/JFM-item?26.0081.01
http://www.emis.de/cgi-bin/JFM-item?28.0061.08

I think these were the papers that were later translated
into English by P. Jourdain and published in 1915 (and
later reprinted by Dover) as "Contributions to the
Founding of the Theory of Transfinite Numbers".
I thought I had once come across a digital copy of
"Contributions" on the internet somewhere, but I
can't find one right now.

Dave L. Renfro

[K. P. Hart]

Dave Seaman wrote:

>>>According to a previous thread in sci.math, Cantor was the first to
>>>describe what is now called the Cantor set in terms of base-3
>>>representations. This establishes a natural bijection between a subset
>>>of the real line and the power set of the naturals. I don't know the
>>>exact history, but it seems likely that this observation may have helped
>>>lead to the second diagonal proof, the one involving decimal digits.
>>>
>>>
>>>
>>>
>>That's my point: he did *not* use decimals in the diagonal argument;
>>follow the link that I supplied: ``we take two symbols m and w and consider
>>the set of all sequences of these symbols ...''
>>I want to know who converted Cantor's `m' and `w' into decimals in order
>>to bring the proof back to the reals, so to speak.
>>
>>
>
>No, *you* need to read what *I* wrote. I didn't say he used decimals; I
>said he used *base-3* representations in proving that the Cantor set is
>uncountable. A real number x in [0,1] belongs to the Cantor set iff it
>has a base-3 representation that uses only the digits 0 and 2.
>
>There is an obvious bijection between the set of all characteristic
>functions on the naturals and the set of all mappings f: N -> {0,2}.
>
>Moving from base 3 to base 10 does not seem like such a huge leap to me.

>http://uk.geocities.com/fr...@btinternet.com/cantor/diagarg.htm
>
>
>
I read what you wrote and I don't dispute that moving from one base to
the other
is not such a big leap but what I find surprizing/amazing/astounding is
that very many
authors refer to the paper that I linked to
(http://uk.geocities.com/fr...@btinternet.com/cantor/diagarg.htm)
and then, in one way or another, proceed to `quote' from it the
diagonal-argument-with-decimals.
Even though
1. Cantor does nothing in base 10 in that paper
2. His purpose is not to show that R is incountable but to give a proof
that uncountable magnitudes exist that avoids working with irrational
numbers.

I do not dispute that Cantor *may* have thought of the
base-10-diagonalization; the problem
is that I can't find it in his works, so I'd like to know who first
wrote it down (one respondend mentioned Borel and I intend to pursue
that reference).

KP

--
E-MAIL: K.P....@EWI.TUDelft.NL PAPER: Faculteit EWI
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URL: http://fa.its.tudelft.nl/~hart 2600 GA Delft
the Netherlands

[K. P. Hart]

Achava Nakhash, the Loving Snake wrote:

>K. P. Hart wrote:
>
>
>>Contrary to popular belief Cantor's (original) diagonal argument did not
>>involve decimal
>>expansions of real numbers, see, e.g.,
>>http://uk.geocities.com/fr...@btinternet.com/cantor/diagarg.htm
>>Also his stated purpose was not to demonstrate the uncountability of the
>>real line
>>but to show that there are larger magnitudes than the countable,
>>*without* involving
>>irrational numbers.
>>
>>What I am interested in is learning who actually first diagonalized
>>decimal expansions.
>>Any pointers are appreciated.
>>
>>K. P. Hart
>>
>>
>
>A most interesting article. Cantor maintains that the uncountability
>of the real numbers in any finite interval follows from this proof, and
>
>

Actually, no.
The paper linked to above has no Section 2; he means that the proof
in section 2 of the earlier paper (``from the proof in section 2 *there*创)
establishes at once that an interval (a,b) is not countable.

[Dave Seaman]

I thought your question was about who first formulated the base-10
argument, not where it might be found in that particular paper.

> I do not dispute that Cantor *may* have thought of the
> base-10-diagonalization; the problem
> is that I can't find it in his works, so I'd like to know who first
> wrote it down (one respondend mentioned Borel and I intend to pursue
> that reference).

As I said, it was established in a previous thread in sci.math that
Cantor used the base-3 definition of what is now called the Cantor set.
That is, he considered strings of base-3 digits. I don't have the
reference at hand right now, but he was surely aware that this
constituted a proof of the uncountability of the reals, based on their
representations in some base.

[Geevarghese Philip]

But the set of all sequences of two symbols _is_ (a subset of) the reals
expressed in binary, so your question is only who converted from the
binary to the decimals -- the `real' angle was already there with Cantor's
original proof, right?

If that is the case, the conversion from binary to decimal might have been
done for pedagogical reasons, just as a didactic device, by one of the
first people who tried to explain Cantor's proof to a college class,
perhaps?

Regards,
Philip

[Dave L. Renfro]

K. P. Hart wrote (in part):

>> What I am interested in is learning who actually
>> first diagonalized decimal expansions. Any pointers
>> are appreciated.

Dave L. Renfro wrote (in part):

> I'm pretty sure the decimal diagonal argument is in
> Borel's 1898 text "Leçons sur la Théorie des Fonctions",
> but my (photo) copy of it is at home so I can't check on

> this right now.I looked last night and it appears that Borel

I looked last night and it didn't appear that the
decimal diagonal argument is in Borel's 1898 book,
but I can only make out bits and pieces of French
and I didn't spend a lot of time looking up words
and trying to follow what some of the non-symbolic
text material said. It appeared that Borel used a
continued fraction representation for the reals
to establish something (this avoids the problems
with non-uniqueness of decimal expansions), but
I wasn't quite sure what.

There's also Lebesgue's 1904 book, which is at

http://name.umdl.umich.edu/ACM0062.0001.001

but when I looked at it briefly just now, I didn't
see what you're looking for. However, you should
probably look at it yourself and spend more time
than I did, which was about three minutes (much
of which was spent waiting for .pdf file downloads).

If you have access to a research library, you should
go look at the old volumes of C. R. Acad. Sci. Paris
from around 1895 to 1905. Look in the table of contents
(in the back of the volumes?) for where the math articles
are, and quickly glance at them. This shouldn't take
you more than an hour at the most if the volumes are
accessible to you on shelves. You can also get .pdf
files for the articles (via the JFM web pages), but
I don't think there's a way to search specifically
for papers published in C. R. Acad. Sci. Paris, so
you'd have to look them up by author or title, and
thus risk missing some. The reason I suggest C. R.
Acad. Sci. Paris is because this is where people
used to announce their results before publishing
full accounts or to publish less profound comments
and observations, and so there's a good chance,
if the explicit decimal argument is due to a French
mathematician, that it would have first appeared
here.

Better yet, just ask someone who should know,
such as José Ferreirós, who is a regular contributor
in the Historia Mathematica list:

http://www.pdipas.us.es/j/josef/

http://mathforum.org/kb/forum.jspa?forumID=149

Dave L. Renfro