First diagnal proof for real numbers
WM
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
Keith Ramsay
The diagonal proof was not Cantor's first proof of the
uncountability of the reals. The first proof constructed a
nested sequence of intervals, the n-th of which does not
contain the first n terms of the given sequence of reals,
whose lengths converge to zero, and hence whose intersection
is a real not in the sequence. This proof doesn't suffer
from the same difficulty.
Keith Ramsay
Dave L. Renfro
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.
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.
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
Dave L. Renfro
WM
> On May 11, 4:39 am, WM <mueck...@rz.fh-augsburg.de> 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
>
> The diagonal proof was not Cantor's first proof of the
> uncountability of the reals.
Yes. Therefore I am in doubt whether Cantor considered his 1891 proof
as a proof concerning the real numbers or not. If he implicitly meant
to include the real numbers, then his proof was in error (with respect
to neglecting the identity of 1.000... and 0.999...). In fact he used
a binary system with symbols "w" and "m" (most probably for German
"female" and "male").
Although Cantor referred to his 1874 proof in the first line of his
1891 proof, I am not sure about his intention. Perhaps there are other
sources known to other mathematicians. For quick reference, I include
the first paragraph of Cantor's 1891 paper in my translation and, for
comparison, in the original version:
In my paper with the heading: About a property of the epitome of all
real algebraic numbers (Journ. Math. vol. 77, p. 258) a proof is
given, probably for the first time, that there are infinite manifolds
which cannot be put in a unique relation with the totality of all
finite integers 1, 2, 3, ..., nu, ..., or, how I use to say, these
manifolds have not the richness / extent [meant but not said is
"cardinality"] of the series of numbers 1, 2, 3, ..., nu, ... [today
we say "sequence", but Cantor uses "series" in all his writings].
>From that what was proven in § 2 we can conclude immediately that,
e.g., the totality of all real numbers cannot be represented in form
of a series omega_1, omega_2, ..., omega_nu, .... [omega is used as a
variable in Cantor's 1874 paper. As a symbol denotig the countably
infinite, Cantor introduced omega not before 1883.]
Now, there is a much simpler proof of this theorem which is
independent of the consideration of irrational numbers.
"In dem Aufsatze, betitelt: Über eine Eigenschaft des Inbegriffs aller
reellen algebraischen Zahlen (Journ. Math. Bd. 77, S. 258), findet
sich wohl zum ersten Male ein Beweis für den Satz, daß es unendliche
Mannigfaltigkeiten gibt, die sich nicht gegenseitig eindeutig auf die
Gesamtheit aller endlichen ganzen Zahlen 1, 2, 3, ..., nu, ...
beziehen lassen, oder, wie ich mich auszudrücken pflege, die nicht die
Mächtigkeit der Zahlenreihe 1, 2, 3, ..., nu, ... haben. Aus dem in §
2 Bewiesenen folgt nämlich ohne weiteres, daß beispielsweise die
Gesamtheit aller reellen Zahlen eines beliebigen Intervalles
(alpha ... beta) sich nicht in der Reihenform omega_1, omega_2, ...,
omega_nu, ... darstellen läßt.
Es läßt sich aber von jenem Satze ein viel einfacherer Beweis liefern,
der unabhängig von der Betrachtung der Irrationalzahlen ist."
> The first proof constructed a
> nested sequence of intervals, the n-th of which does not
> contain the first n terms of the given sequence of reals,
> whose lengths converge to zero, and hence whose intersection
> is a real not in the sequence. This proof doesn't suffer
> from the same difficulty.
That is correct. Thank you very much. But what is your opinion
concerning Cantor's intention?
Regards, WM
WM
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?
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
Dave L. Renfro
>> 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.
WM wrote:
> 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?
Borel (1898) gives Cantor's 1874 proof (pp. 14-15) and
Cantor's 1892 proof (pp. 107-108). In the case of Cantor's
1892 proof, Borel first gives a proof that is very similar
to what Cantor gave, except Borel formulates the proof as
the uncountability of all functions taking only the values
0 and 1 (and defined, I think, on an arbitrary set; or at
least on an arbitrary set of real numbers, which allows
for the two important special cases, the natural numbers
and an interval). Following this, on p. 108, Borel mentions
that this shows the uncountability of all numbers between
0 and 1 expressed in binary "decimal" form, and then Borel
remarks:
"Ils reconnaîtront, en particulier, que le cas où, à partir
d'un certain rang, tous les chiffres décimaux seraient égaux
à 1, donne lieu à une difficulté aisée à écarter."
(google's translation) "They will recognize, in particular,
that the case where, starting from a certain row, all the
decimal digits would be equal to 1, place gives to a difficulty
easy to draw aside."
My feeling is that despite Cantor not explicitly mentioning
the non-uniqueness of expansions, he was well aware of it.
In Joseph Dauben's 1979 book about Cantor, and in the translations
of the letters between Cantor and Dedekind in William Ewald's
1996 book "From Kant to Hilbert: A Source Book in the Foundations
of Mathematics", there is quite a bit about Cantor spending a
few days (a week?) worrying about non-uniqueness of decimal
expansions (pointed out by Dedekind, I think) when Cantor was
proving that R, R^2, R^3, etc. all have the same cardinality
(by interlacing decimal expansions). This was around 1877
or 1878 I think. Cantor resolved the problem in a rather
constructive way. He avoided the more abstract method of
"taking a countable set away from an uncountable set leaves
the cardinality of the uncountable set unchanged" (which I'm
sure he considered, but perhaps he thought this involved less
certain reasoning) by using continued fraction expansions,
which real numbers have unique expansions in terms of.
Also, Cantor started out in number theory. His Ph.D. and
first few papers were in number theory, and in an 1872
paper on trigonometric series expansions [the one that
could be considered the birth of set theory, because he
introduces first species sets (= sets with a finite
Cantor-Bendixson rank) in this paper], Cantor devoted
the first part of his paper to constructing the real
numbers (essentially via what is now known as the method
of constructing the reals from the rationals by the use
of Cauchy sequences of rational numbers, with an appropriate
equivalence relation imposed). Then there's the "Cantor
expansion" (which you can google) of a real number,
which I assume is due to Georg Cantor and not the math
historian Moritz Cantor (but I'm not certain right now).
These expansions are unique, although I don't know when
they first appeared in the literature. This too argues in
favor of Cantor being fully aware of the implications of
his 1892 proof as a way to show the uncountability
of the reals and the issue that needs to be taken care
of if decimal expansions are used.
Also, continued fraction expansions were much more a
part of the typical mathematician's background during
the early 1890's period than is the case now. I think
the problem with non-uniqueness of decimal expansions
would have been far less an issue then than now. My
feeling, one that I've slowly come to realize over the
past few years (because I too have been interested in
the same types of questions you're asking about), is that
our present day preoccupation with whether mathematicians
then were aware of the problems of non-uniqueness of decimal
expansions in the diagonal proof is due to our seeing
the past through our present filter/lens, where much of
what used to be general knowledge about continued fraction
expansions has been filtered out.
Dave L. Renfro wrote:
>> 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.
I learned this from José Ferreirós's 1999 book
"Labyrinth of Thought: A History of Set Theory
and its Role in Modern Mathematics" (top of p. 306),
who writes:
"It has to be noted that it was Russell, not Cantor in his
published work, who focused on the Cantor Theorem as a
central result of great importance. He seems to have been
the first mathematician who presented it as showing that
the set of all subsets of S has always a greater cardinality
than S itself [Russell (1903), Sections 346-347]. Thus,
it was Russell who formulated it for the first time as a
purely set-theoretical result. (In Cantor's version it
showed that, given a set S, a certain set of _functions_
has greater cardinality, and functions were _not_ taken
to be sets.) In the process, Russell was the first to
emphasize something like the Power Set Axiom. All of
this was in itself an important contribution, for until
then the theorem lay rather forgotten in the first annual
report of the DMV and its significance had not been
clearly grasped."
However, I think Ferreirós may have overlooked Borel's
1898 book in this regard. After getting a copy of this
book (from a local university library) to more carefully
look over the part about Cantor's diagonal proof,
which I had previously seen and thus knew was in Borel's
book, I see that Borel also calls attention to the
connection between Cantor's 1892 proof and the collection
of all subsets of a set. In the last section of his
"Note I: La Notion Des Puissances" (pp. 102-110),
titled "La puissance des ensembles de fonctions"
(pp. 109-110), Borel clearly introduces the notion
of a characteristic function (but doesn't name the
notion -- this was due to Vallee-Poussin in 1915,
I think, who also made strong use of the idea in
measure theory) and shows how Cantor's proof implies
that the collection of all subsets of the reals has
cardinality greater than the set of reals.
Dave L. Renfro
Dave L. Renfro
> 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 I need to make some more corrections.
Both addition and multiplication of cardinals are
defined, and some of their arithmetic properties
mentioned (but not investigated very much), in Cantor's
1887 paper "Mitteilungen zur Lehre vom Transfiniten".
I think ordinal exponentiation might be discussed in
this paper also, but I'm not sure. I do know that Cantor
discussed the idea of an epsilon number (an ordinal that
is a solution to the ordinal exponential equation w^x = x)
in an 11 October 1886 letter to Franz Goldscheider. The
letter is in Dauben's 1979 book (Appendix, pp. 304-306;
the letter is in German and is not translated in Dauben's
book). Without knowing any German, it appears to me that
Cantor may simply be working formally, without a precise
definition of ordinal exponentiation in mind, but this
is just a guess on my part.
Back to the original poster's question, the following
is from p. 96 of the English translation (by Jourdain)
of Cantor's 1895 and 1897 papers, "Contributions to the
Founding of the Theory of Transfinite Numbers", Dover
Publications, 1915/1952:
----------------------------
If we denote the power of the linear continuum X
(that is, the totality X of real numbers x such
that x >= and <= 1) by c [my symbol, not Cantor/Jourdain's,
which is non-ascii), we easily see that it may be
represented by, amongst others, the formula
(11) c = 2^(aleph_0),
where [section] 6 gives the meaning of aleph_0. In fact,
by (4), 2^(aleph_0) is the power of all representations
(12) x = f(1)/2 + f(2)/2^2 + ... + f(v)/2^v + ...
of the numbers x in the binary system. If we pay
attention to the fact that every number x is only
represented once, with the exception of the numbers
x = (2v + 1)/2^u < 1, which are represented twice
over, we have, if we denote the "enumerable" totality
of the latter by {s_v},
2^(aleph_0) = card({s_v}, X).
If we take away from X any "enumerable" aggregate
{t_v} and denote the remainder by X_1, we have
[snip some equations and expressions I assume demonstrate
the claim]
----------------------------
WM wrote (in part, in another post):
> 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://tinyurl.com/2zzmfh [tinyurl equivalent]
It looks to me that this is simply an oversight on Whitehead
and Russell's part, not something that people in general
weren't aware of. (Although I'm very surprised an oversight
such as this would be in their work.) On p. 64 they write
"If the nth figure [digit] in the nth decimal is p, let the nth
figure in N be p+1 (or 0, if p=9). Then N is different from all
the members [elements] of E, since ..."
Even if we just look at representations only involving
numbers strictly between 0 and 1, there will be a problem.
For the list
0.49999999...
0.49111111...
0.44911111...
0.44491111...
0.44449111...
. . . . . . .
there will be a problem, since their procedure leads to
the number 0.500000..., which is equal to the first number
on the list. The same thing can be done with any number
that has two decimal representations, since one of the
representations for any such number will have a tail
of 9's.
Dave L. Renfro
WM
On 13 Mai, 18:47, "Dave L. Renfro" <renfr...@cmich.edu> wrote:
> Dave L. Renfro wrote:
> >> 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.
[...]
> "Ils reconnaîtront, en particulier, que le cas où, à partir
> d'un certain rang, tous les chiffres décimaux seraient égaux
> à 1, donne lieu à une difficulté aisée à écarter."
So we have now a fine final result concerning the first recognition of
the special substitution rule necessary for Cantor's diagonal proof
using n-ary representations of real numbers:
Borel (1898) probably was the first to mention it.
Jules Richard (1905) seems to have been the second. (Small wonder, as
a French mathematician he might have read Borel's book. Unfortunately
there is next to nothing known about the life of Richard. All
published biographical data seem to be taken from J. Itard: ''Richard,
Jules Antoine'', Dictionary of Scientific Biography, '''11''', Charles
Scribner's Sons, New York (1980) 413-414. I tried to get some more
information, for a Wikipedia article,
http://en.wikipedia.org/wiki/Jules_Richard
from the French lycées where he had taught --- but in vain.)
Also Fraenkel in his "Einleitung in die Mengenlehre" (at least in the
second edition, Springer 1923) explains a rule in detail (never use 0
for replacement).
Zermelo, as the editor of Cantor's collected works 1932, also mentions
the problem in a footnote.
Cantor, living until 1918, not publishing after 1897 though but
continuing to work on set theory according to a report by Kowalewski
(Bestand und Wandel, München 1950) did never come back to his proof
(as far as I could see from his collected works and his
correspondence, most of which has been published by Meschkowski and
Nilson Springer, Berlin 1991).
Whitehead and Russell 1910 in PM overlook the problem, although
already Russell in 1903, contrary to Cantor, explicitly considers
numbers as an example: "The two characters m and w may be considered
respectively as greater and less than some fixed term. Thus the x's
may be rational numbers [...] These remarks are logically irrelevant,
but they make the argument easier to follow." And they contain an
error. (It is always great for a small man to find a small error in a
great man's work)
[...]
Dave L. Renfro wrote:
> My feeling is that despite Cantor not explicitly mentioning
> the non-uniqueness of expansions, he was well aware of it.
> In Joseph Dauben's 1979 book about Cantor, and in the translations
> of the letters between Cantor and Dedekind in William Ewald's
> 1996 book "From Kant to Hilbert: A Source Book in the Foundations
> of Mathematics", there is quite a bit about Cantor spending a
> few days (a week?) worrying about non-uniqueness of decimal
> expansions (pointed out by Dedekind, I think) when Cantor was
> proving that R, R^2, R^3, etc. all have the same cardinality
> (by interlacing decimal expansions). This was around 1877
> or 1878 I think.
Cantor's letter with the decimal expansion is dated June 20, 1877.
Dedekind responds two days later, outlining the problem. Cantor
accepts immediately (postcard of June 23). On June 25 he informs
Dedekind about the continued fractions method, which had been the
original version of his proof, worked out before the decimal version
already. (But the continued fractions method is somewhat more
complicated. So he probably wanted to simplify his proof.) Already on
June 29 he pushes Dedekind to react. This letter contains the famous
phrase "je le vois, mais je ne crois pas." Dedekind congratulates on
July, 2, but utters his doubts concerning the continuity of the
mapping (proof of discontinuity by Brouwer in 1911).
>
> Also, Cantor started out in number theory. His Ph.D. and
> first few papers were in number theory, and in an 1872
> paper on trigonometric series expansions [the one that
> could be considered the birth of set theory, because he
> introduces first species sets (= sets with a finite
> Cantor-Bendixson rank) in this paper], Cantor devoted
> the first part of his paper to constructing the real
> numbers (essentially via what is now known as the method
> of constructing the reals from the rationals by the use
> of Cauchy sequences of rational numbers, with an appropriate
> equivalence relation imposed). Then there's the "Cantor
> expansion" (which you can google) of a real number,
> which I assume is due to Georg Cantor and not the math
> historian Moritz Cantor (but I'm not certain right now).
Georg Cantor's theory of irrational numbers, which he later called
"Fundamentalreihen" (fundamental series) is published for the first
time in "Ueber die Ausdehnung eines Satzes aus der Theorie der
trigonometrischen Reihen, Math. Annalen vol. 5, pp. 123 - 132 (1872)
> These expansions are unique, although I don't know when
> they first appeared in the literature. This too argues in
> favor of Cantor being fully aware of the implications of
> his 1892 proof as a way to show the uncountability
> of the reals and the issue that needs to be taken care
> of if decimal expansions are used.
There can be no question that Cantor was fully aware of the double
decimal expansion. Nevertheless, he did not mention it. Although he
emphasizes in the final section of his 1891 paper that his diagonal
proof can be generalized, and explicitly includes real functions, he
seems to have been convinced that the replacement of one digit per
sequence in any case is sufficient to distinguish them:
"so sieht man ohne weiteres, daß die Gleichung E_0 = E_mu für keinen
positiven ganzzahligen Wert von mu erfüllt sein kann da sonst für das
betreffende mu und für alle ganzzahligen Werte von nu b_nu = a_mu,nu,
also auch im besonderen b_mu = a_mu,mu wäre, was durch Definition von
b_nu ausgeschlossen ist."
My translation:... so one sees immediately that the equation E_0 =
E_mu (the E_mu denote the entry sequences, E_0 is the constructed
diagonal) for no positive integer mu can be satisfied because
otherwise for that mu and for all integer values of nu b_nu = a_mu,nu,
in particular b_mu = a_mu,mu which can be excluded by the definition
of b_nu. (The a_mu and b_nu are the elements of the sequences and of
the diagonal, repectively.)
In fact the problem of Cantor's proof with double decimal expansion is
not present for sequences of m's and w's. Considering only such
sequences, we have, e.g.,
mwww... =/= wmmm...
and Cantor's statement is fully correct.
> Dave L. Renfro wrote:
> >> 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.
>
> I learned this from José Ferreirós's 1999 book
> "Labyrinth of Thought: A History of Set Theory
> and its Role in Modern Mathematics" (top of p. 306),
>
> who writes:
>
> "It has to be noted that it was Russell, not Cantor in his
> published work, who focused on the Cantor Theorem as a
> central result of great importance. He seems to have been
> the first mathematician who presented it as showing that
> the set of all subsets of S has always a greater cardinality
> than S itself [Russell (1903), Sections 346-347]. Thus,
> it was Russell who formulated it for the first time as a
> purely set-theoretical result. (In Cantor's version it
> showed that, given a set S, a certain set of _functions_
> has greater cardinality, and functions were _not_ taken
> to be sets.)
Here I would not agree. Cantor considers the set of real functions
f(x) which take only the values 0 and 1 for x between 0 and 1
(collected works p. 279) and concludes on even higher cardinal
numbers. Further Cantor introduced the "Belegungsfunktion" which
obviously represents a set (Georg Cantor, Gesammelte Abhandlungen, E.
Zermelo (ed.), Springer 1932, p. 287).
___________________________
Dave L. Renfro wrote in the next contribution of May, 14:
>
> It seems I need to make some more corrections.
> Both addition and multiplication of cardinals are
> defined, and some of their arithmetic properties
> mentioned (but not investigated very much), in Cantor's
> 1887 paper "Mitteilungen zur Lehre vom Transfiniten".
> I think ordinal exponentiation might be discussed in
> this paper also, but I'm not sure. I do know that Cantor
> discussed the idea of an epsilon number (an ordinal that
> is a solution to the ordinal exponential equation w^x = x)
> in an 11 October 1886 letter to Franz Goldscheider. The
> letter is in Dauben's 1979 book (Appendix, pp. 304-306;
> the letter is in German and is not translated in Dauben's
> book). Without knowing any German, it appears to me that
> Cantor may simply be working formally, without a precise
> definition of ordinal exponentiation in mind, but this
> is just a guess on my part.
Cantor never was a formalist, and he would have been rather sad about
the later development of set theory. In his letter to Goldscheider he
uses complete induction to get the sequence of epsilon numbers (there
he uses the character gamma)
g_1 = Li (w, w_1, w_2, ...)
g_1, g_2, ..., g_w, ...
with w_1 = w^w, w_2 = w^w_1 etc.
(g = gamma, w = omega)
This letter is also included in Meschkowski's and Nilsson's: "Geog
Cantor, Briefe", Springer, 1991. The editors write (I translate only
the most important remarks, and those not literally):
After the catastrophe of 1885(*) there was no mathematician of
distinction with whom Cantor could discuss his ideas. So the interest
of Franz Goldscheider, a gymnasium (secondary school) teacher and
former pupil of Cantor, was very welcome to him. In a series of some
very long letters, but also in personal conversation, Cantor explained
his theory to him. ... While the first letter of June 18, 1886 can be
considered some introduction to set theory, later the contents quickly
becomes more demanding. Much of what he published 10 years later in
his "Beiträge" is anticipated here. This letter is a convincing
example. It is very similar to § 20 (of the Beiträge) about the
epsilon numbers - here called giants - of the second number class.
(*) Cantor had to withdraw one of his papers from Acta Mathematica
when it had reached the proof stage because, according to Mittag-
Leffler, it came about one hundred years too soon.
> ----------------------------
>
> WM wrote (in part, in another post):
>
> > Whitehead and Russell 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://tinyurl.com/2zzmfh [tinyurl equivalent]
>
> It looks to me that this is simply an oversight on Whitehead
> and Russell's part, not something that people in general
> weren't aware of.
This problem is not present for sequences of digits. It merely results
from the limit
{i --> oo} n^-i = 0
which is required to guarantee convergence of the n-ary expansion of
real numbers.
In the definition of the real numbers there are two antagonistic
principles, namely digits and negative powers, a_i and 10^-i,
respectively:
SUM {i = 1 to oo} a_i * 10^-i
The negative powers are required to guarantee convergence of the n-ary
expansion of real numbers. If two expansions have different a_i and
b_i for only one index i e N, then the two numbers are different,
unless the limit for n --> oo spoils this distinction. But Cantor's
original diagonal proof concerns only the part a_i =/= b_i and does
not concern, at least it does not consider, limits at all. So the
problem of 0.100... = 0.0999... may easily be overlooked.
> (Although I'm very surprised an oversight
> such as this would be in their work.) On p. 64 they write
>
> "If the nth figure [digit] in the nth decimal is p, let the nth
> figure in N be p+1 (or 0, if p=9). Then N is different from all
> the members [elements] of E, since ..."
>
> Even if we just look at representations only involving
> numbers strictly between 0 and 1, there will be a problem.
>
> For the list
>
> 0.49999999...
> 0.49111111...
> 0.44911111...
> 0.44491111...
> 0.44449111...
> . . . . . . .
>
> there will be a problem, since their procedure leads to
> the number 0.500000..., which is equal to the first number
> on the list. The same thing can be done with any number
> that has two decimal representations, since one of the
> representations for any such number will have a tail
> of 9's.
It should be noted that the convergence of the decimal expansion also
implies the limit
lim {i --> oo} (a_i - b_i) n^-i = 0.
Regards, WM
Dave L. Renfro
> So we have now a fine final result concerning the
> first recognition of the special substitution rule
> necessary for Cantor's diagonal proof using n-ary
> representations of real numbers:
>
> Borel (1898) probably was the first to mention it.
While working on something else (but related), I came
across two earlier appearances in the literature that
are explicit about this issue.
---------------------------------------------------------
Felix Klein, "Vorträge über ausgewählte Fragen der
Elementargeometrie", B. G. Teubner, 1895, v + 66 pages.
[Preface dated Ostern (Easter) 1895. JFM 26.0546.01]
http://quod.lib.umich.edu/cgi/b/bib/bibperm?q1=ACV2370.0001.001
Charlotte Angas Scott, Review of Felix Klein's "Vorträge
über ausgewählte Fragen der Elementargeometrie", Bulletin
of the American Mathematical Society (2) 2 (1895-96), 157-164.
The following is from pp. 162-163 of C. A. Scott's review,
which discusses what can be found on p. 42 of Klein's book.
The next step is to show how numbers can be
constructed that shall not be contained in this
orderly series. The number being required to lie
within certain limits, so that there are given
a certain number of decimal places, e.g., 5, the
digits in the following places have to be selected
so that the number differs from all of the series.
For a reason explained, the digit 9 is avoided.
The 6th digit is chosen to be different from the
6th of the first algebraic number, and thus the
number constructed will certainly be different
from this; the 7th digit is chosen to be different
from the 7th of the second algebraic number, by
which we ensure that the number written down is
not the second, and so on. Hence we are assured
of the existence of numbers that are not the roots
of any algebraic equation; that is, the existence
of transcendental numbers is proved, and it is shown
how they can be written down. Moreover, since the
choice of the digit to be written in any assigned
place is restricted only by the exclusion of two
digits, 9 and one other, we may choose any one
of the 8 that are left, zero being admissible in
this same way as any other. Hence between any two
algebraic numbers there are 8^oo transcendental
numbers, (not oo^8 as stated in the pamphlet,)
[the last 2 commas appear exactly as I've placed
them] and real algebraic numbers form only a small
part of all numbers.
---------------------------------------------------------
Heinrich Weber, "Lehrbuch der Algebra", Volume II,
Friedrich Vieweg und Sohn, 1896, xiv + 796 pages.
[Preface dated July 1896. JFM 27.0056.01]
http://dz1.gdz-cms.de/no_cache/dms/load/toc/?IDDOC=45274
Heinrich Weber, "Transcendental numbers", translation by
Wooster Woodruff Beman of Chapter 25 (pp. 745-767) of
Heinrich Weber's "Lehrbuch der Algebra" (Volume II),
Bulletin of the American Mathematical Society (2)
3 (1896-97), 174-195. [JFM 28.0084.01]
Pages 750-751 of Weber's book (Volume II) gives the
diagonal proof specifically for decimal expansions of
real numbers. There is a lengthy review of Weber's book
by James Pierpont in Bull. Amer. Math. Soc. (2) 4
(1897-98), 200-234, but as far as I could tell there
was no mention of either of Cantor's proofs in it.
The following is from pp. 178-179 of W. W. Beman's
translation:
This theorem may be demonstrated in another way
which is simpler in some respects and may be
briefly indicated. [The earlier proof was Cantor's
1874 proof.] We do not restrict the generality if
we confine ourselves to the interval from 0 to 1.
We shall imagine all numbers of this interval
represented by decimal fractions with an infinite
number of terms. Finite decimal fractions are
included if we make all the digits after a certain
one equal to zero. To render this representation by
decimal fractions unambiguous, it must be agreed
that for a finite decimal fraction this representation
must _always_ be chosen, so that, for example,
0.4999 ... must not be written for 0.5000 ...
We will now assume that these decimal fractions
form an enumerable mass. They may then be arranged
in a countable series, represented as follows:
[replace 'a' with '\alpha' and 'b' with '\beta']
\omega_1 = 0.a_{1}^{(1)}a_{2}^{(1)}a_{3}^{(1)} . . .
\omega_2 = 0.a_{1}^{(2)}a_{2}^{(2)}a_{3}^{(2)} . . .
\omega_3 = 0.a_{1}^{(3)}a_{2}^{(3)}a_{3}^{(3)} . . .
. . . . . . . . . . . . . . . . . . . . .
where the a_{\mu}^{(\nu)} represent digits of the
decimal system.
But it is very easy to form a decimal fraction (or
indeed, as many as we please) which is not contained
in the series \Omega. We have only to form
\eta = 0.b_1b_2b_3 . . .
where the b_{\nu} are digits of the decimal system,
satisfying the one condition that for every \nu,
b_{\nu} is different from a_{\nu}^{(\nu)}. This
number \eta, which also belongs to the interval
(0,1) cannot be a number of the series \Omega.
The formation of \eta may be made still more
general by arbitrarily selecting the b's as far
as we please and _then_ applying the law,
b_{\nu} >< a_{\nu}^{(\nu)}.
---------------------------------------------------------
Dave L. Renfro
WM
part):
> Felix Klein, "Vorträge über ausgewählte Fragen der
> Elementargeometrie", B. G. Teubner, 1895, v + 66 pages.
> [Preface dated Ostern (Easter) 1895. JFM 26.0546.01]http://quod.lib.umich.edu/cgi/b/bib/bibperm?q1=ACV2370.0001.001
>
> Heinrich Weber, "Lehrbuch der Algebra", Volume II,
> Friedrich Vieweg und Sohn, 1896, xiv + 796 pages.
> [Preface dated July 1896. JFM 27.0056.01]http://dz1.gdz-cms.de/no_cache/dms/load/toc/?IDDOC=45274
Thank you, Dave, for your scrutiny. So we have now, as a final result,
that it lasted only few years from Cantor's diagonal proof for general
sequences, published in 1892, till its application to real numbers
together with the correct statement concerning the dual representation
of some rationals.
Felix Klein in 1895
and Heinrich Weber in 1896
predate Emile Borel in 1898
and Jules Richard in 1905.
Further it appears as if Heinrich Weber was the first to give a
"Cantor-matrix" in the form persisting in modern text books.
Regards, WM