On the cardinal of the Cantor set
José Carlos Santos
The Cantor set has the same cardinal as the set of real numbers. A way
of proving this fact is this one: a number x belongs to the Cantor set
iff x can be written in base 3 as 0,a_1a_2a_3a_4..., where each a_n is
either 0 or 2. Then, to each such x we can associate the real number
which, in base 2, can be written as 0,(a_1/2)(a_2/2)(a_3/2)... This
defines a surjective function from C onto [0,1].
My question is: does anyone know who is the author of this proof?
Best regards,
Jose Carlos Santos
Dave Seaman
> The Cantor set has the same cardinal as the set of real numbers. A way
> of proving this fact is this one: a number x belongs to the Cantor set
> iff x can be written in base 3 as 0,a_1a_2a_3a_4..., where each a_n is
> either 0 or 2. Then, to each such x we can associate the real number
> which, in base 2, can be written as 0,(a_1/2)(a_2/2)(a_3/2)... This
> defines a surjective function from C onto [0,1].
But it's not a bijection. What you actually get is a bijection between the
Cantor set and P(N), the power set of the integers.
> My question is: does anyone know who is the author of this proof?
Cantor certainly knew that |R| = |P(N)|. One of the most obvious ways of
proving that is to use the Cantor-Bernstein theorem.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
José Carlos Santos
>>The Cantor set has the same cardinal as the set of real numbers. A way
>>of proving this fact is this one: a number x belongs to the Cantor set
>>iff x can be written in base 3 as 0,a_1a_2a_3a_4..., where each a_n is
>>either 0 or 2. Then, to each such x we can associate the real number
>>which, in base 2, can be written as 0,(a_1/2)(a_2/2)(a_3/2)... This
>>defines a surjective function from C onto [0,1].
>
> But it's not a bijection. What you actually get is a bijection between the
> Cantor set and P(N), the power set of the integers.
I know it's not a bijection! But since there's a surjectiove function
from the Cantor set onto [0,1], the cardinal of the Cantor set is grater
or equal to the cadinal of the set of real numbers. Since, on the other
hand, the Cantor set is a set of real numbers, it follows that both sets
have the same cardinal.
>>My question is: does anyone know who is the author of this proof?
>
>
> Cantor certainly knew that |R| = |P(N)|. One of the most obvious ways of
> proving that is to use the Cantor-Bernstein theorem.
I have explicitly asked who was the author of *that* proof, not who
discovered that theorem.
Best regards,
JOse Carlos Santos
Mike Oliver
> Dave Seaman wrote:
JCS> The Cantor set has the same cardinal as the set of real numbers. A way
JCS> of proving this fact is this one: a number x belongs to the Cantor set
JCS> iff x can be written in base 3 as 0,a_1a_2a_3a_4..., where each a_n is
JCS> either 0 or 2. Then, to each such x we can associate the real number
JCS> which, in base 2, can be written as 0,(a_1/2)(a_2/2)(a_3/2)... This
JCS> defines a surjective function from C onto [0,1].
>>
>> But it's not a bijection. What you actually get is a bijection
>> between the
>> Cantor set and P(N), the power set of the integers.
>
>
> I know it's not a bijection! But since there's a surjectiove function
> from the Cantor set onto [0,1], the cardinal of the Cantor set is grater
> or equal to the cadinal of the set of real numbers. Since, on the other
> hand, the Cantor set is a set of real numbers, it follows that both sets
> have the same cardinal.
So this is a valid argument, but you might notice that it
can't be formalized without AC or some fragment thereof.
That means it leaves open the possibility that the cardinalities
might not be the same in some otherwise-reasonable model of ZF
that we actually care about (say L(R)).
But in fact there's no such possibility, because you can
get *injections* both ways between the Cantor set
and [0,1], and then apply Schroeder-Bernstein.
Dave Seaman
> Dave Seaman wrote:
>>>The Cantor set has the same cardinal as the set of real numbers. A way
>>>of proving this fact is this one: a number x belongs to the Cantor set
>>>iff x can be written in base 3 as 0,a_1a_2a_3a_4..., where each a_n is
>>>either 0 or 2. Then, to each such x we can associate the real number
>>>which, in base 2, can be written as 0,(a_1/2)(a_2/2)(a_3/2)... This
>>>defines a surjective function from C onto [0,1].
>> But it's not a bijection. What you actually get is a bijection between the
>> Cantor set and P(N), the power set of the integers.
> I know it's not a bijection! But since there's a surjectiove function
> from the Cantor set onto [0,1], the cardinal of the Cantor set is grater
> or equal to the cadinal of the set of real numbers. Since, on the other
> hand, the Cantor set is a set of real numbers, it follows that both sets
> have the same cardinal.
Do you know what the Cantor-Bernstein theorem says? If you do, then why
did you just repeat what I had already said?
>>>My question is: does anyone know who is the author of this proof?
>> Cantor certainly knew that |R| = |P(N)|. One of the most obvious ways of
>> proving that is to use the Cantor-Bernstein theorem.
^^^^^^^^^^^^^^^^^^^^^^^^
> I have explicitly asked who was the author of *that* proof, not who
> discovered that theorem.
What proof? What you described in your original question was not a proof
at all. I made it into a proof by invoking the Cantor-Bernstein theorem.
If your question is who was the first to describe the bijection between
P(N) and 2^N (one being the set of all subsets of N and the other being
the set of all characteristic functions on N), I think you will find that
it is implicit in Cantor's definition of A^B for cardinal numbers A and
B.
José Carlos Santos
>>>>The Cantor set has the same cardinal as the set of real numbers. A way
>>>>of proving this fact is this one: a number x belongs to the Cantor set
>>>>iff x can be written in base 3 as 0,a_1a_2a_3a_4..., where each a_n is
>>>>either 0 or 2. Then, to each such x we can associate the real number
>>>>which, in base 2, can be written as 0,(a_1/2)(a_2/2)(a_3/2)... This
>>>>defines a surjective function from C onto [0,1].
>
>
>>>But it's not a bijection. What you actually get is a bijection between the
>>>Cantor set and P(N), the power set of the integers.
>
>
>>I know it's not a bijection! But since there's a surjective function
>>from the Cantor set onto [0,1], the cardinal of the Cantor set is grater
>>or equal to the cardinal of the set of real numbers. Since, on the other
>>hand, the Cantor set is a set of real numbers, it follows that both sets
>>have the same cardinal.
>
>
> Do you know what the Cantor-Bernstein theorem says? If you do, then why
> did you just repeat what I had already said?
Yes, I know know what it says, and I do not feel that I was repeating
what you said.
>>>>My question is: does anyone know who is the author of this proof?
>
>
>
>>>Cantor certainly knew that |R| = |P(N)|. One of the most obvious ways of
>>>proving that is to use the Cantor-Bernstein theorem.
>
> ^^^^^^^^^^^^^^^^^^^^^^^^
>
>
>>I have explicitly asked who was the author of *that* proof, not who
>>discovered that theorem.
>
>
> What proof? What you described in your original question was not a proof
> at all. I made it into a proof by invoking the Cantor-Bernstein theorem.
I posted a partial proof, because I thought that anyone would understand
how to complete it.
> If your question is who was the first to describe the bijection between
> P(N) and 2^N (one being the set of all subsets of N and the other being
> the set of all characteristic functions on N), I think you will find that
> it is implicit in Cantor's definition of A^B for cardinal numbers A and
> B.
My question has already been clearly stated. Read the first line of
this post below "On 03-03-2005 23:52, Dave Seaman wrote:"; it says
"The Cantor set has the same cardinal as the set of real numbers." This
is the statement that I am interested in. And the proof whose author I
would like to know is the one that uses the fact that every element of
the Cantor set can be written in base 3 using only the digits 0 and 2
together with the fact that the elements of of [0,1] are precisely those
real numbers that can be written in base 2 in the form 0,a1a2a3a4...
Now, I've answered your question, although all of this was already
contained in my first post.
Best regards,
Jose Carlos Santos.
José Carlos Santos
> So this is a valid argument, but you might notice that it
> can't be formalized without AC or some fragment thereof.
> That means it leaves open the possibility that the cardinalities
> might not be the same in some otherwise-reasonable model of ZF
> that we actually care about (say L(R)).
>
> But in fact there's no such possibility, because you can
> get *injections* both ways between the Cantor set
> and [0,1], and then apply Schroeder-Bernstein.
Indeed, that's a better way of doing it.
Dave Seaman
> On 03-03-2005 23:52, Dave Seaman wrote:
>>>>>The Cantor set has the same cardinal as the set of real numbers. A way
>>>>>of proving this fact is this one: a number x belongs to the Cantor set
>>>>>iff x can be written in base 3 as 0,a_1a_2a_3a_4..., where each a_n is
>>>>>either 0 or 2. Then, to each such x we can associate the real number
>>>>>which, in base 2, can be written as 0,(a_1/2)(a_2/2)(a_3/2)... This
>>>>>defines a surjective function from C onto [0,1].
>>>>But it's not a bijection. What you actually get is a bijection between the
>>>>Cantor set and P(N), the power set of the integers.
>>>I know it's not a bijection! But since there's a surjective function
>>>from the Cantor set onto [0,1], the cardinal of the Cantor set is grater
>>>or equal to the cardinal of the set of real numbers. Since, on the other
>>>hand, the Cantor set is a set of real numbers, it follows that both sets
>>>have the same cardinal.
>> Do you know what the Cantor-Bernstein theorem says? If you do, then why
>> did you just repeat what I had already said?
> Yes, I know know what it says, and I do not feel that I was repeating
> what you said.
No, you used surjections instead of the injections that are mentioned in
the Cantor-Bernstein theorem. But, as I have mentioned elsewhere, Cantor
would have considered it to be the same argument, since he implicitly
assumed that every set can be well ordered.
>>>>>My question is: does anyone know who is the author of this proof?
>>>>Cantor certainly knew that |R| = |P(N)|. One of the most obvious ways of
>>>>proving that is to use the Cantor-Bernstein theorem.
>> ^^^^^^^^^^^^^^^^^^^^^^^^
>>>I have explicitly asked who was the author of *that* proof, not who
>>>discovered that theorem.
>> What proof? What you described in your original question was not a proof
>> at all. I made it into a proof by invoking the Cantor-Bernstein theorem.
> I posted a partial proof, because I thought that anyone would understand
> how to complete it.
Yes, and I did exactly that. Two different ways, in fact. One way uses
the Cantor-Bernstein theorem, and the other doesn't. I'll explain more
in a moment. But that raises the question: which proof, exactly, did
you have in mind?
>> If your question is who was the first to describe the bijection between
>> P(N) and 2^N (one being the set of all subsets of N and the other being
>> the set of all characteristic functions on N), I think you will find that
>> it is implicit in Cantor's definition of A^B for cardinal numbers A and
>> B.
See the explanation below concerning why this is relevant to the question.
> My question has already been clearly stated. Read the first line of
> this post below "On 03-03-2005 23:52, Dave Seaman wrote:"; it says
> "The Cantor set has the same cardinal as the set of real numbers." This
> is the statement that I am interested in. And the proof whose author I
> would like to know is the one that uses the fact that every element of
> the Cantor set can be written in base 3 using only the digits 0 and 2
> together with the fact that the elements of of [0,1] are precisely those
> real numbers that can be written in base 2 in the form 0,a1a2a3a4...
> Now, I've answered your question, although all of this was already
> contained in my first post.
I see two ways to complete the argument you presented and make it a
proof. One, as I said, is to use C-B. You seem to object to the
presence of that theorem in the proof. Very well, here is a method that
doesn't use that theorem.
The argument that you described gives us a bijection between the Cantor
set and the set of all infinite binary sequences. This is the set that I
call 2^N, or the set of all characteristic functions on N. Cantor's
theorem shows that |2^x| > |x| for every set x. Hence, 2^N is
uncountable and therefore so is the Cantor set.
So your question comes down to, who first recognized that |{0,2}^N| =
|{0,1}^N|? That is, the set of {0,2}-sequences has the same cardinality
as the set of {0,1}-sequences? I think you will find that the answer is
Cantor, because it is implicit in his definition of cardinal
exponentiation. If |A| = |A'| and |B| = |B'|, then |A^B| = |A'^B'|.
Dave L. Renfro
Well, it looks like my earlier post/rant got through
the new google beta-groups, so I guess I'll go ahead
and chime in here as well.
I looked in a few places but I didn't find the answer.
I might look harder at a later time. If I find the answer,
I'll come back to this thread and post it.
However, my main reason for jumping in is to engage in
a mini-rant about math history. (This must be my day
for rants.)
I've found that trying to determine the answer to
even the most basic questions (such as yours) is
often extremely difficult. Part of the problem is
that sometimes different people worked out different
aspects of something that someone else at a later
time integrated together, or something else like
this that tends to blur the issue of who is
responsible for what. But the biggest problem
for me is not this, but the fact that mathematicians
tend to be very careless with these things. I wish
more people would put some care into what they say
in their published papers, and I'm not talking
about the math, but rather the historical remarks.
If you don't know who did something, or you "think"
someone did something, the best thing is to not say
anything at all or be upfront with what you know and
what you don't know. For example, recently I've been
working on some issues concerning the size of the
set of Liouville numbers and its complement, and I
can't begin to tell you how many times I've read
that in 1844 Liouville published the well-known
theorem concerning how well algebraic numbers of
degree n can be approximated by rational numbers.
No, no, no! Liouville might have known about this
in 1844, but the result itself didn't appear in
print until Liouville's 1851 paper on transcendental
numbers. Liouville's 1844 transcendental proofs
involved numbers given as continued fractions,
and it just so happens that continued fractions
provide very efficient rational approximations,
but at that time (at least in print) Liouville
had not yet identified the more general approximation
property in our present definition of the Liouville
numbers as the fundamental principle at work in
his proofs.
Besides this, I've come across another example within
the past couple of weeks. At the top of p. 105 of
Michael Hallett's "Cantorian Set Theory and Limitation
of Size" is the sentence "In his [Hausdorff's 1914
book] he introduced the first classification according
to complexity of Borel sets." Again, no, no, no!
Lebesgue did this in 1905. (What Hausdorff did, among
many other things that there's no need to get into at
this point, was introduce the F_sigma, G_delta, etc.
notation.) Granted, Hallett isn't a mathematician,
but he is a mathematical historian which makes the
matter even worse. And it's not as if Lebesgue's 1905
paper is all that obscure, since this is the paper
that Lebesgue made his famous error in that Souslin
discovered, which led to the "discovery" of analytic
sets and the birth of all the activity by Polish and
Russian mathematicians in the 1920's in what is now
called descriptive set theory. In fact, Hallett even
gives a fairly decent survey of the development of this
field (up to the 1980's) in the next 10 or 11 pages
of his book! I'd also be willing to bet that Hausdorff,
in the 1914 edition of his book that Hallett refers
to (which I don't have a copy of, but Hausdorff's
1935 3'rd edition certainly mentions Lebesgue's 1905
paper -- see "Further References" for Section 43),
explicitly says that this is due to Lebesgue.
Dave L. Renfro
José Carlos Santos
>>Yes, I know know what it says, and I do not feel that I was repeating
>>what you said.
>
> No, you used surjections instead of the injections that are mentioned in
> the Cantor-Bernstein theorem. But, as I have mentioned elsewhere, Cantor
> would have considered it to be the same argument, since he implicitly
> assumed that every set can be well ordered.
I didn't know that. It's interesteing.
> Yes, and I did exactly that. Two different ways, in fact. One way uses
> the Cantor-Bernstein theorem, and the other doesn't. I'll explain more
> in a moment. But that raises the question: which proof, exactly, did
> you have in mind?
I am not interested in how to complete the proof, since what I'm
interested in is that part of the proof that uses the expression of
real numbers in base 2 and base 3.
> So your question comes down to, who first recognized that |{0,2}^N| =
> |{0,1}^N|? That is, the set of {0,2}-sequences has the same cardinality
> as the set of {0,1}-sequences?
No, my question is: who was the first person who used expressing real
numbers in the bases 2 and 3 in order to prove that the Cantor set has
the same cardinal as the set of real numbers.
Best regards,
Jose Carlos Santos
Dave Seaman
> Dave Seaman wrote:
> No, my question is: who was the first person who used expressing real
> numbers in the bases 2 and 3 in order to prove that the Cantor set has
> the same cardinal as the set of real numbers.
From what I have seen, Cantor considered perfect aggregates, but he did
not give any special attention to the middle-thirds version of that set.
Dave L. Renfro
> From what I have seen, Cantor considered perfect aggregates,
> but he did not give any special attention to the middle-thirds
> version of that set.
I think p. 590 of the following paper is where Cantor
first introduces the middle-thirds set -->
enter '590' in the window under "Go to Page"
and then click on "Go to Page"
http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D26429
Also, I believe somewhere around p. 479 of the next paper
Cantor shows the set has measure zero. (Not Lebesgue measure,
but I think the measure he's working with is equivalent to
Lebesgue measure for compact sets.)
http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D26526
I say "I think", because I can't read a bit of German,
and these papers are in German. One would think with
all the interest in Cantor that someone by now would have
posted English translations on the internet. This relates
back to my mini-rant yesterday on being sloppy with history
in math publications. Because I can pretty much only read
English, I'm reduced to looking at secondary sources
when I'm trying to find out who did something [1]. Since
mathematicians are often sloppy with this (as I explained
yesterday), I often come across inconsistencies that can
be very time consuming to resolve. It's usually not much
trouble to get the original paper(s), as I can drive to
a good research library or use interlibrary loan. But
knowing *exactly* what the author says and does not say
is quite another matter.
[1] Something I find especially irksome is when an author
of a historical paper includes a large number of quotes
in the original German, French, etc. in an English
paper without providing a translation. I find myself
throwing up my hands in frustration, because if I
could read the original language I would have been
reading the original to begin with. A. F. Monna is
especially bad with this. I have a biography of
Hermann Hankel [Nieuw Archief voor Wiskunde (3) 21
(1973), 64-87] that is almost useless because
everytime Monna is about to say something specific
that's not in several other secondary sources I
have, Monna launches into a lengthy quote from
Hankel, Cantor, Abel, Du Bois-Reymond, or Gauss.
(Regarding my interest in Hankel, google the phrase
"Hermann Hankel" in UseNet.)
Dave L. Renfro
G. A. Edgar
Dave L. Renfro <renf...@cmich.edu> wrote:
> I think p. 590 of the following paper is where Cantor
> first introduces the middle-thirds set -->
p. 590 does, indeed, describe the set with digits 0 and 2
base 3.
> I say "I think", because I can't read a bit of German,
> and these papers are in German. One would think with
> all the interest in Cantor that someone by now would have
> posted English translations on the internet.
My book CLASSICS ON FRACTALS
<http://www.amazon.com/exec/obidos/tg/detail/-/0813341531/>
has an English translation of a different
1884 paper of Cantor, where he proves all uncountable closed sets
in R have power c, and of course he uses Cantor sets to do it
(but of course not just "middle-thirds" Cantor sets).
> Because I can pretty much only read
> English, I'm reduced to looking at secondary sources
Back in the Olden Days when I was in graduate school, we had
to show that we could read mathematics in 2 foreign languages.
So your remark means that when our graduate students pressure us
to eliminate the foreign
language requirement for the Ph.D. (saying it is useless
nowadays), they are wrong?
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Lee Rudolph
...
>Back in the Olden Days when I was in graduate school, we had
>to show that we could read mathematics in 2 foreign languages.
In the same Olden Days (albeit at the other end of Mass. Ave.),
Frank Peterson defined the foreign language requirement as
"being able to get a good hotel and order a good meal" in (two?)
foreign languages. Reading wasn't explicitly mentioned.
About 20 years later, I mentioned that definition to Ronnie Lee,
who proposed adding "brothel" to the list.
Lee Rudolph
Stephen J. Herschkorn
you should take the time to learn French and German, at least.
Mathmematical French is not very hard to read once you know the basics.
You can then decide if you want to learn Russian, Latin, and Italian.
Languages are as much prerequisite to (math) history as are basic
analysis, topology, and algebra are to higher mathematics.
--
Stephen J. Herschkorn sjher...@netscape.net
Dave Seaman
> Dave Seaman wrote:
>> From what I have seen, Cantor considered perfect aggregates,
>> but he did not give any special attention to the middle-thirds
>> version of that set.
> I think p. 590 of the following paper is where Cantor
> first introduces the middle-thirds set -->
> enter '590' in the window under "Go to Page"
> and then click on "Go to Page"
> http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D26429
Thanks for the reference. Cantor does indeed introduce the base-3
representation with all digits 0 or 2 in the middle of that page, which I
think answers the original question.
The World Wide Wade
<050320051643280014%ed...@math.ohio-state.edu.invalid>,
"G. A. Edgar" <ed...@math.ohio-state.edu.invalid> wrote:
> > Because I can pretty much only read
> > English, I'm reduced to looking at secondary sources
>
> Back in the Olden Days when I was in graduate school, we had
> to show that we could read mathematics in 2 foreign languages.
>
> So your remark means that when our graduate students pressure us
> to eliminate the foreign
> language requirement for the Ph.D. (saying it is useless
> nowadays), they are wrong?
They are wrong to claim it is useless. They are right to say it
is nearly useless.
Andrea
I don't know. But I do know that is possible to prove the theorem
without using numbers written in base 3 or 2.
Maybe the real question is who discovered that (in fact) the Cantor
set consist exactly in numbers that are able to be written in such
form in base 3.
José Carlos Santos
> Maybe the real question is who discovered that (in fact) the Cantor
> set consist exactly in numbers that are able to be written in such
> form in base 3.
Cantor himself, since that's what he used as the *definition* of the
Cantor set.
José Carlos Santos
> I think p. 590 of the following paper is where Cantor
> first introduces the middle-thirds set -->
That's confirmed by Thomas Hawkins in his book "Lebesgue's Theory of
Integration: Its Origins and Development".
Dave L. Renfro
> That's confirmed by Thomas Hawkins in his
> book "Lebesgue's Theory of Integration:
> Its Origins and Development".
I did a little digging around today when I
probably should have been working on something
else, and I uncovered a few more things.
In the following, Borel shows that the set of
numbers of the form
SUM(n=1 to oo) of k_n / 10^(n!),
where k_n are digits 0, 1, ..., 9, has cardinality c
by matching each such number with the decimal
expansion of a number between 0 and 1,
SUM(n=1 to oo) of k_n / 10^(n).
(Borel also deals with the issue of rational
numbers having two different representations.)
Borel (1898 text, pp. 28-29)
http://www.emis.de/cgi-bin/JFM-item?29.0336.01
Much of Borel's 1898 book (above) is an expansion
of his 1894 Ph.D. Thesis, published in 1895 (below),
but it doesn't seem to me as if the argument above
appears anywhere in it.
Borel (1895, Ann. Sci. l'Ècole Norm. Sup., pp. 9-55)
http://www.numdam.org/numdam-bin/feuilleter?id=ASENS_1895_3_12_
The following deals with the Cantor middle thirds set,
but I'm not sure whether anything relevant to your
question is discussed.
Brodén (1901, Math Ann., p. 519)
http://gdz-srv3.sub.uni-goettingen.de/sub/digbib/loader?did=D36412
The first edition of Lebesgue's integration theory
book (below) discusses some aspects of the Cantor
middle thirds set.
Lebesgue (1904 book, pp. 26-27)
http://name.umdl.umich.edu/ACM0062
Lebesgue's book (above) was a slight revision
of this Ph.D. Thesis, published in 1902, but
I don't know if the Cantor set issues in his
1904 book appear in this earlier work.
Lebesgue (1902, Annali di Mat., pp. 231-359)
http://www.emis.de/cgi-bin/JFM-item?33.0307.02
The following quote indicates that Cantor and
others were probably in possession by the mid
1880's of the proof you're interested in, but if
you want to know the first _explicit_ appearance
of this proof, it might be Borel's 1898 book
(although you still have to read between the lines).
Julian F. Fleron, "A note on the history of the
Cantor set and Cantor function", Mathematics
Magazine 67 (1994), 136-140.
"During the time Cantor was working on the
'Punktmannichfaltigkeiten' papers, others
were working on extensions of the Fundamental
Theorem of Calculus to discontinuous functions.
Cantor addressed this issue in a letter [18]
dated November 1883, in which he defines the
Cantor set, just as it was defined in the
paper [14] of 1883 (which had actually been
written in October in 1882). However, in the
letter he goes on to defined the Cantor function,
the first known appearance of this of this
function. It is first defined on the complement
of the Cantor set to be the function whose
values are
(1/2)*[ c_1/2 + ... + c_{n-1}/2^{n-1} + 2/2^n ]
for any number between
a = c_1/3 + ... + c_{n-1}/3^{n-1} + 1/3^n
and
b = c_1/3 + ... + c_{n-1}/3^{n-1} + 2/3^n,
where each c_k is 0 or 2. Cantor then concludes
this section of the letter by noting that this
function can be extended naturally to a continuous
increasing function on [0,1]. That serves as a
counterexample to Harnack's extension of the
Fundamental Theorem of Calculus to discontinuous
functions, which was in vogue at the time (see
e.g. [2, p. 60]). We are given no indication
of how Cantor came upon this function."
(Reference [2] is Hawkins' book.)
Finally, the following paper contains a lot of
information. I haven't looked at it in a while,
so I don't know what it says regarding your
present question.
Akihiro Kanamori, "The mathematical development of
set theory from Cantor to Cohen", Bulletin of Symbolic
Logic 2 (1996), 1-71.
http://www.math.ucla.edu/~asl/bsl/0201-toc.htm
Dave L. Renfro
Dave L. Renfro
> Seems to me that if you really want to be able to study
> math history, you should take the time to learn French
> and German, at least. Mathmematical French is not very
> hard to read once you know the basics. You can then
> decide if you want to learn Russian, Latin, and Italian.
>
> Languages are as much prerequisite to (math) history as
> are basic analysis, topology, and algebra are to higher
> mathematics.
I agree, but unfortunately I've had a lot of difficulty
with languages all my life. I'm one of those rare cases
of someone who has a knack for math that isn't very good
with Foreign languages. I was extremely lucky to be in a
graduate program that required only one language and lucky
again that I managed to pass the language exam (French).
I wasn't so lucky as an undergraduate, when my problems
caused two delays in graduation, the second being when
I transferred to a less selective university after several
of my attempts at language competency were so far from
acceptable that I didn't see any other alternative.
My interest in math history is as an amateur. Mainly,
I just like to know how something I'm interested in
developed, and in most cases this doesn't take me much
before 1900. However, 1900 seems to be where most
professional math historians stop but earlier than
where brief surveys in research papers begin, so I
often find myself having to dig up a lot of the stuff
that's in-between.
Dave L. Renfro
Dave L. Renfro
> My book CLASSICS ON FRACTALS
> <http://www.amazon.com/exec/obidos/tg/detail/-/0813341531/>
> has an English translation of a different 1884 paper
> of Cantor, where he proves all uncountable closed sets
> in R have power c, and of course he uses Cantor sets to
> do it (but of course not just "middle-thirds" Cantor sets).
Yes, I know about this book, and I assume you know
that I know [1] and that you're only mentioning this for
the original poster (who should also see [2] below).
[1] You might recall my telling you what a great service
books like this are at the June 1994 Charlottesville
conference. The most significant thing about this
book for me is the English translation of Hausdorff's
1918 paper where Hausdorff dimension was introduced
(and quite thoroughly investigated, even in this
first paper).
[2] William Ewald (editor), "From Kant to Hilbert:
A Source Book in the Foundations of Mathematics",
Clarendon Press, 1999.
Volume 2 contains quite a few translations of
letters by Cantor and and others, as well as a
translation of Cantor's 1874 paper (pp. 840-843)
where Cantor first proved the uncountability of R
by a nested intervals argument that generalizes
easily to the Baire category theorem.
> Back in the Olden Days when I was in graduate school,
> we had to show that we could read mathematics in 2
> foreign languages.
Two languages was the norm even in the late 1980's,
but I guess that's not the case anymore.
> So your remark means that when our graduate students
> pressure us to eliminate the foreign language
> requirement for the Ph.D. (saying it is useless
> nowadays), they are wrong?
I think it should be maintained, but not because
they're going to need it as much as me. However,
given the relative ease that most people in math
have with languages, I don't see it as much of a
hurdle. In fact, I think if the student is given
a particularly important work in their field
to translate, it could be quite helpful to both
the student and to others. On my shelves is an
English translation of Zygmunt Zahorski, "Sur la
primiere derivee", Transactions of the American
Mathematical Society 69 (1950), 1-54 that James
Foran did for one of his graduate school language
competencies. I also have an English translation
of the famous "Five letters on set theory" that
appeared in 1905 (Hadamard, Borel, Lebesgue, and
Baire) which a student of Dan Mauldin's did as part
of their language competency (neither of them at
the time realized that van Heijenoort's 1967 book
has these translated). I think these sorts of
things should be done far more often.
Dave L. Renfro
José Carlos Santos
> I did a little digging around today when I
> probably should have been working on something
> else, and I uncovered a few more things.
Your contributions to this thread have been outstanding. Thanks a lot!
Jose Carlos Santos
Shmuel (Seymour J.) Metz
at 05:06 PM, lrud...@panix.com (Lee Rudolph) said:
>In the same Olden Days (albeit at the other end of Mass. Ave.), Frank
>Peterson defined the foreign language requirement as "being able to
>get a good hotel and order a good meal" in (two?) foreign languages.
>Reading wasn't explicitly mentioned.
>About 20 years later, I mentioned that definition to Ronnie Lee, who
>proposed adding "brothel" to the list.
I'm glad that my department didn't have such a requirement. I could
read Mathematics papers and texts in French, but I doubt that I would
have been able to find a good brothel, hotel or restaurant in French.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
Dave L. Renfro
> The first edition of Lebesgue's integration theory
> book (below) discusses some aspects of the Cantor
> middle thirds set.
>
> Lebesgue (1904 book, pp. 26-27)
> http://name.umdl.umich.edu/ACM0062
>
> Lebesgue's book (above) was a slight revision
> of this Ph.D. Thesis, published in 1902, but
> I don't know if the Cantor set issues in his
> 1904 book appear in this earlier work.
>
> Lebesgue (1902, Annali di Mat., pp. 231-359)
> http://www.emis.de/cgi-bin/JFMitem?33.0307.02
Yesterday I got the first three volumes of
Lebesgue's OEUVRES SCIENTIFIQUES (his "collected
works", published in 1972) from our interlibrary
loan office for something I've been working on
(not the issues discussed below).
Lebesgue's 1902 Thesis is in Volume I, so I'm now
able to say something about it. The Cantor middle
thirds set is briefly discussed on pp. 10-11
(= pp. 212-213 of O-S, Vol. I), the page numbering
being for his actual Thesis (as opposed to its
unaltered -- I would assume -- publication in
Annali di Mat.). Lebesgue shows that it has measure
zero and states that it has cardinality of the
continuum. Lebesgue concludes that there exist
measurable sets that are not Borel sets (Borel
sets are called "measurable (B) sets" here).
This is the standard result that follows from
the fact that each of the 2^c subsets of the
Cantor set has measure zero, and hence are
measurable, whereas there are only c many Borel
sets. This last result can be proved rigorously
using transfinite induction, and I believe
Lebesgue does this in his 1905 paper "Sur les
fonctions représentables analytiquement", but
my impression is that this is one of those things
that everyone "knew to be true", but no one had
yet done so in an entirely rigorous way. (This
is just a feeling I have without having researched
the matter.) In any event, on the previous page
(p. 10), just before introducing the Cantor
middle thirds set, Lebesgue writes:
"Ceux que l'on peut obtenir par cette méthode
et leurs complémentaires sont ceux que M. Borel
appelle mesurables [footnote here to pp. 46 and
50 of Borel's 1898 book] et que nous nommerons
ensembles mesurables (B) [this term is italized].
Ils sont définis par une infinité dénombrable
de conditions, leur ensemble a la puissance
du continu."
Here Lebesgue is introducing the term "measurable (B)
sets" for the collection of sets that Borel defined
(differently than is done today, by the way) and then
Lebesgue observes that since they are defined through
the use of a countably infinite number of conditions,
there are c many of them. I don't know off-hand whether
anyone before this argued that there are c many of them,
but I would guess that Borel did so in his 1898 book
(and probably also Baire in his 1899 Thesis).
While I'm at it, I may as well mention that on
p. 12 Lebesgue also points out that there exist
"ensembles mesurables (J)" (Jordan measurable sets)
that are not Borel using the same argument. (Every
subset of the Cantor middle thirds set is also
Jordan measurable.)
Regarding Lebesgue's 1904 book, I previously
mentioned that the Cantor middle thirds set is
discussed briefly on pp. 26-27. Here, when
Lebesgue says that this set has cardinality c,
he gives the following footnote for one
justification. ('Z' is the Cantor middle thirds set.)
"On peut dire aussi que Z a la puissance du continu
parce qu'il dépend d'une infinité dénombrable de
constantes entières a_1, a_2, ...."
Rough translation: "We can also say that Z has the
power of the continuum because it depends on a
countable infinity of natural numbers a_1, a_2, ...."
The a_n's are the digits of the ternary expansion
of real numbers belonging to the Cantor set.
The reason for the "also" aspect in Lebesgue's
footnote is that in the text itself Lebesgue
observes that Z is a perfect set (nonempty
obviously intended), and hence has cardinality c.
Dave L. Renfro
Dave L. Renfro
>> My book CLASSICS ON FRACTALS
>> <http://www.amazon.com/exec/obidos/tg/detail/-/0813341531/>
>> has an English translation of a different 1884 paper
>> of Cantor, where he proves all uncountable closed sets
>> in R have power c, and of course he uses Cantor sets to
>> do it (but of course not just "middle-thirds" Cantor sets).
Dave L. Renfro wrote (in part):
> Yes, I know about this book, and I assume you know
> that I know [1] and that you're only mentioning this for
> the original poster (who should also see [2] below).
>
> [1] You might recall my telling you what a great service
> books like this are at the June 1994 Charlottesville
> conference. The most significant thing about this
> book for me is the English translation of Hausdorff's
> 1918 paper where Hausdorff dimension was introduced
> (and quite thoroughly investigated, even in this
> first paper).
I came across the following comment a couple of days
ago on p. 230 of Janusz Czyz's book "Paradoxes of Measures
and Dimensions Originating in Felix Hausdorff's Ideas",
World Scientific, 1994:
"It is possible that this Hausdorff's above paper is
the most important mathematical publication that has
not been translated in English."
Czyz is talking about Hausdorff's 1918 paper on Hausdorff
dimension. Incidentally, Edgar's book appeared in 1993,
about the same time Czyz's book did.
Dave L. Renfro