Re: First diagnal proof for real numbers

On 11 Mai, 23:57, "Dave L. Renfro" <renfr...@xxxxxxxxx> wrote:
WM wrote:
In his original paper of 1891 G. Cantor does not consider
the fact that dual representation of real numbers can
spoil his diagonal proof. (In fact Cantor treats sequences
in general.) Today it is known that the substitution rule
for real numbers in n-ary representation has to exclude
replacement of n-1 by 0 and vice versa. Although E. Zermelo,
the editor of Cantor's collected works, mentioned this
already in 1932 (G. Cantor, Gesammelte Werke, p. 280-281)
it seems unclear who was the first to give the correct
diagonal proof for real numbers. Regards, WM

There was a problem with Cantor's first attempt (in letters
to Dedekind) to prove R^2 and R have the same cardinality.
Cantor switched from decimal expansions to continued fraction
expansions to overcome the non-uniqueness issues with
decimal expansions. However, I don't think there was
a problem with his 1892 paper (talk given in September 1891,
publication date of the paper is 1892). Cantor simply
showed the uncountability of all sequences of elements
from a two element set. If the goal was to give a
simple proof that uncountable sets exist, then there's
no problem. Near the beginning Cantor writes (translated):
"But it is possible to give a much simpler proof of that
theorem which does not depend on considering the irrational
numbers." It is not clear from this and from his preceding
statements (well, it's not clear to me) whether "that
theorem" is the existence of uncountable sets or the
more specific result that the set of real numbers is
uncountable.

Exactly this point is also unclear to me.

Incidentally, Cantor's original paper is available
on the internet in digital form:

http://dz1.gdz-cms.de/index.php?id=toc&no_cache=1&IDDOC=243972

The link at this page for Cantor's paper actually
takes you to the paper just before Cantor's paper.
However, if you enter '75' for the page selection,
you'll be taken to Cantor's paper.

Borel's 1898 book has some discussion of Cantor's
diagonal proof near the end, and he might be the first
person to explicitly deal with decimal (or n-ary)
expansions of real numbers and diagonalization.

Thank you. So this might be the first mention of the precaution to be
taken in order to avoid the problem of 0.999... = 1.000... in the
diagonal proof?

For those who might be interested . . .

When Cantor's 1892 diagonalization paper was
published, Cantor had not yet formulated cardinal
exponentiation. Moreover, Cantor did not explicitly
phrase his result in terms of the collection of
all subsets of a set.

Before 1895, no arithmetic operations had been defined
for cardinal numbers and only addition and multiplication
had been defined for ordinal numbers. (The addition
and multiplication of ordinal numbers was first given
in Cantor's 1883 "Grundlagen" work, published separately
and as part 5 of his "Punktmannichfaltigkeiten" series
of papers.) Cantor formulated cardinal exponentiation
in 1895 and it appears in Section 4 (pp. 486-488) of
his "Beitrage" (Part I). Also, although ordinal
exponential notational symbols such as w^w, w^(n*w),
w^w^w, etc. appeared in his early 1880's papers
(even in his early-mid 1870's papers, but with oo instead
of w being used), Cantor did not formulate the operation
of exponentiation for ordinal numbers until Part II of
his "Beitrage" (1897).

It seems that the first person to make an explicit
connection between Cantor's 1892 diagonalization
result and the collection of all subsets of a set
was Bertrand Russell in 1903. See Sections 346-347
(pp. 364-366) of Russell's book, which is also on
the internet in digital form, at:

http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAT1273

In this connection it is interesting to note that Jules Richard
mentioned the correct rule when using Cantor's diagonal proof for
constructing his paradox (which is not an antinomy):

We form a number having zero for the integral part and p + 1 for the n-
th decimal, if p is not equal either to 8 or 9, and unity in the
contrary case.
[The principles of mathematics and the problem of sets (1905), English
translation in Jean van Heijenoort, "From Frege to Gödel - A Source
Book in Mathematical Logic", 1879-1931. Harvard Univ. Press, 1967, p.
142-144.]

Whitehead and Russel in Principia Mathematica (p. 64), however,
skipped or forgot this precaution: Here only the digit 9 is replaced
by the digit 0, such that identities like 1.000... = 0.999... can
spoil the result.

Principia Mathematica, available at the U. Michigan. The link leads to
pp. 59-78:
http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=AAT3201.0001.001;didno=AAT3201.0001.001;view=pdf;seq=00000083

Regards, WM



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Relevant Pages

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