Re: A consideration concerning the diagonal argument of G. Cantor
WM
> In article <1bbcdd80-41f2-407a-91f9-2676d14f3...@34g2000hsh.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
> ...
> > But as I already mentioned much earlier. Cantor's argument fails,
> > because it must cover any conceivable case. One simple counter-
> > argument spoiling it is the sequence
> >
> > 0.0
> > 0.1
> > 0.11
> > 0.111
> > ...
> >
> > and the replacement rule 0 --> 1 which yields an anti-diagonal that is
> > in the sequence, as /every/ initial segment can be found in the list.
>
> Every *finite* initial segment is in the list.
How do you get from finity to infinity in a static (not dynamic) set?
If the anti-diagonal, has aleph_0 digits, then there is at least one
list entry with aleph_0 - 1 digits, by construction. How much is that?
The *actual* , static existence of an infinity of digits is in
contradiction with mathematics. This shows that transfinite set theory
is nonsense - one of the biggest mistakes of human intellectual
history. It could even be considered the biggest one unless set theory
was so absolutely meaningless and insignificant.
Regards, WM
Virgil
<13520f43-72c4-4880...@p49g2000hsd.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 3 Okt., 14:11, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > In article
> > <1bbcdd80-41f2-407a-91f9-2676d14f3...@34g2000hsh.googlegroups.com> WM
> > <mueck...@rz.fh-augsburg.de> writes:
> > ...
> > > But as I already mentioned much earlier. Cantor's argument fails,
> > > because it must cover any conceivable case. One simple counter-
> > > argument spoiling it is the sequence
> > >
> > > 0.0
> > > 0.1
> > > 0.11
> > > 0.111
> > > ...
> > >
> > > and the replacement rule 0 --> 1 which yields an anti-diagonal that is
> > > in the sequence, as /every/ initial segment can be found in the list.
> >
> > Every *finite* initial segment is in the list.
>
> How do you get from finity to infinity in a static (not dynamic) set?
You don't, no set is dynamic, you need a different set.
>
> If the anti-diagonal, has aleph_0 digits, then there is at least one
> list entry with aleph_0 - 1 digits, by construction.
What do you man by aleph_0 -1? The most reasonable meaning requires that
aleph_0 - n = aleph_0, for any natural n.
Dik T. Winter
> On 3 Okt., 14:11, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
...
> > > and the replacement rule 0 --> 1 which yields an anti-diagonal that is
> > > in the sequence, as /every/ initial segment can be found in the list.
> >
> > Every *finite* initial segment is in the list.
>
> How do you get from finity to infinity in a static (not dynamic) set?
Not.
> If the anti-diagonal, has aleph_0 digits, then there is at least one
> list entry with aleph_0 - 1 digits, by construction. How much is that?
Not so. I see no construction there.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
WM
> In article <13520f43-72c4-4880-ab6a-5d59e8d4c...@p49g2000hsd.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 3 Okt., 14:11, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> ...
> > > > and the replacement rule 0 --> 1 which yields an anti-diagonal that is
> > > > in the sequence, as /every/ initial segment can be found in the list.
> > >
> > > Every *finite* initial segment is in the list.
> >
> > How do you get from finity to infinity in a static (not dynamic) set?
>
> Not.
>
> > If the anti-diagonal, has aleph_0 digits, then there is at least one
> > list entry with aleph_0 - 1 digits, by construction. How much is that?
>
> Not so. I see no construction there.
1
11
111
...
You can't see that there is no diagonal with X elements unless there
is at least one line with X elements? Sorry, then we are discussing in
different worlds.
Regards, WM
Daryl McCullough
>1
>11
>111
>...
>
>You can't see that there is no diagonal with X elements unless there
>is at least one line with X elements?
If X is a natural number, then that's true. It's not true for the
case X = omega.
If you want to make a claim of the form "For all X, there is no diagonal
with X elements unless there is at least one line with X elements", then
*PROVE* it. First, you need to say what X ranges over. Does range over
natural numbers? Does it range over all ordinals? What is X?
Then you need to show that your conclusion follows from your
assumptions or axioms. What are your axioms? What are your assumptions?
What are your definitions?
What is your definition of "having X elements"? What does that mean?
>Sorry, then we are discussing in different worlds.
Yes, you are talking about a madeup world of your own invention.
You are pretending to be doing mathematics, when you are actually
just playing around.
--
Daryl McCullough
Ithaca, NY
Herbert Newman
>
> 1
> 11
> 111
> ...
>
> You can't see that there is no diagonal with X elements unless there
> is at least one line with X elements?
>
All *I* see is the following: The number of digits of the diagonal (d) is
_identical_ with the number of lines of the list (L). Now the list has
countable infinitely many (aleph_0-many) lines, hence the diagonal has
countable infinitely many (aleph_0-many) digits, and vice versa. With other
words: the number of lines (of L) = the number of digits (of d) = aleph_0.
Moreover, there's neither a line in L with an "infinite index" nor a digit
in d with an "infinite index" by definition (of lists and sequences). Hence
there's certainly no entry in the list with infinitely many digits, though
the diagonal consists of (countable) infinitely many digits.
>
> Sorry, then we are discussing in different worlds.
>
That's for sure.
Herb
WM
> WM says...
>
> >1
> >11
> >111
> >...
>
> >You can't see that there is no diagonal with X elements unless there
> >is at least one line with X elements?
>
> If X is a natural number, then that's true. It's not true for the
> case X = omega.
It is true for every existing diagonal, because the diagonal does not
exist unless all its elements exist.
>
> If you want to make a claim of the form "For all X, there is no diagonal
> with X elements unless there is at least one line with X elements", then
> *PROVE* it.
The diagonal is defined and constructed that way.
>First, you need to say what X ranges over. Does range over
> natural numbers? Does it range over all ordinals? What is X?
X is any number that can be realized by the elements of a diagonal.
>
> Then you need to show that your conclusion follows from your
> assumptions or axioms. What are your axioms? What are your assumptions?
> What are your definitions?
An element of an infinite set like 111... cannot exist at a position
unless there is a finite sequence reaching to that position. An
infinite sequence does not exist unless all of its elements exist.
> Yes, you are talking about a madeup world of your own invention.
> You are pretending to be doing mathematics, when you are actually
> just playing around.
I am doing mathematics, you are lost in a world of nonsense, believing
in a kind of theology that does not even comfort its adherents.
Regards, WM
Herbert Newman
>
> ... the diagonal does not exist unless all its elements exist.
>
Indeed. A truism.
Herb
Daryl McCullough
>An element of an infinite set like 111... cannot exist at a position
>unless there is a finite sequence reaching to that position. An
>infinite sequence does not exist unless all of its elements exist.
According to what theory of "infinite sequences"? You are using
an incoherent theory of infinite sequences, and you are coming
up with nonsense. That just shows that *you* don't know of a
consistent theory of infinite sequences. Other people have
perfectly consistent theories of infinite sets, but you
refuse to use them, preferring your own inconsistent theories.
>> Yes, you are talking about a madeup world of your own invention.
>> You are pretending to be doing mathematics, when you are actually
>> just playing around.
>
>I am doing mathematics
No, you are pretending.
Herbert Newman
> WM says...
>>
>> I am doing mathematics
>>
> No, you are pretending.
>
I guess one of the many reasons why WM is not able to comprehend
mathematics is the fact that he was never taught real math (i.e. pure
math); and he obviously is not able to teach it to himself (by reading a
good textbook).
Herb
Virgil
<0df634ab-40c2-47f3...@h60g2000hsg.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 6 Okt., 14:34, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > In article
> > <13520f43-72c4-4880-ab6a-5d59e8d4c...@p49g2000hsd.googlegroups.com> WM
> > <mueck...@rz.fh-augsburg.de> writes:
> > > On 3 Okt., 14:11, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > ...
> > > > > and the replacement rule 0 --> 1 which yields an anti-diagonal that
> > is
> > > > > in the sequence, as /every/ initial segment can be found in the
> > list.
> > > >
> > > > Every *finite* initial segment is in the list.
> > >
> > > How do you get from finity to infinity in a static (not dynamic) set?
> >
> > Not.
> >
> > > If the anti-diagonal, has aleph_0 digits, then there is at least one
> > > list entry with aleph_0 - 1 digits, by construction. How much is that?
> >
> > Not so. I see no construction there.
>
> 1
> 11
> 111
> ...
So for which n does the nth entry contain aleph_0 - 1 digits?
>
> You can't see that there is no diagonal with X elements unless there
> is at least one line with X elements? Sorry, then we are discussing in
> different worlds.
If all lines are finite, the anti-diagonal can still be infinite.
At least in any mathematical world which allows a set of al naturals.l
Virgil
<555e2a15-9d4f-4f53...@k7g2000hsd.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 6 Okt., 17:17, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
> > WM says...
> >
> > >1
> > >11
> > >111
> > >...
> >
> > >You can't see that there is no diagonal with X elements unless there
> > >is at least one line with X elements?
> >
> > If X is a natural number, then that's true. It's not true for the
> > case X = omega.
>
> It is true for every existing diagonal, because the diagonal does not
> exist unless all its elements exist.
For the existence of an anti-diagonal, it is only necessary for the nth
line to have n characters, so all lines can be finite while the diagonal
is not.
> >
> > If you want to make a claim of the form "For all X, there is no diagonal
> > with X elements unless there is at least one line with X elements", then
> > *PROVE* it.
>
> The diagonal is defined and constructed that way.
Not in my world.
>
> >First, you need to say what X ranges over. Does range over
> > natural numbers? Does it range over all ordinals? What is X?
>
> X is any number that can be realized by the elements of a diagonal.
According to WM's original statement
"For all X, there is no diagonal with X elements
unless there is at least one line with X elements"
X is the cardinality of the number of characters in the anti-diagonal,
but the index set, N, has a cardinality greater than any of its members,
so that WM is wrong once again.
> >
> > Then you need to show that your conclusion follows from your
> > assumptions or axioms. What are your axioms? What are your assumptions?
> > What are your definitions?
>
> An element of an infinite set like 111... cannot exist at a position
> unless there is a finite sequence reaching to that position. An
> infinite sequence does not exist unless all of its elements exist.
An infinite sequence consists of terms each having a finite index in
that sequence.
Its index set is an infinite set of finite indices, the natural numbers.
Perhaps one day WM will learn to accept that simple truth.
WM
> In article
> <0df634ab-40c2-47f3-a76e-a956ef011...@h60g2000hsg.googlegroups.com>,
>
>
>
>
>
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 6 Okt., 14:34, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > > In article
> > > <13520f43-72c4-4880-ab6a-5d59e8d4c...@p49g2000hsd.googlegroups.com> WM
> > > <mueck...@rz.fh-augsburg.de> writes:
> > > > On 3 Okt., 14:11, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > > ...
> > > > > > and the replacement rule 0 --> 1 which yields an anti-diagonal that
> > > is
> > > > > > in the sequence, as /every/ initial segment can be found in the
> > > list.
> > > > >
> > > > > Every *finite* initial segment is in the list.
> > > >
> > > > How do you get from finity to infinity in a static (not dynamic) set?
>
> > > Not.
>
> > > > If the anti-diagonal, has aleph_0 digits, then there is at least one
> > > > list entry with aleph_0 - 1 digits, by construction. How much is that?
>
> > > Not so. I see no construction there.
>
> > 1
> > 11
> > 111
> > ...
>
> > is at least one line with X elements? Sorry, then we are discussing in
> > different worlds.
>
> If all lines are finite, the anti-diagonal can still be infinite.
Obviously the diagonal 111... cannot be longer than every sequence of
1's in the lines, because the diagonal consists of the ends of these
seqeuences. Where no line ends, there the diagonal cannot exist.
> At least in any mathematical world which allows a set of al naturals
The last sentence contains a self-contradiction.
Regards, WM
Dik T. Winter
> On 6 Okt., 14:34, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > > If the anti-diagonal, has aleph_0 digits, then there is at least one
> > > list entry with aleph_0 - 1 digits, by construction. How much is that?
> >
> > Not so. I see no construction there.
>
> 1
> 11
> 111
> ...
>
> You can't see that there is no diagonal with X elements unless there
> is at least one line with X elements? Sorry, then we are discussing in
> different worlds.
Apparently we are. I am talking mathematics, what are you talking? Anyway
in your list above I still do not see a construction. What I see that each
n-th line has an n-th element which provides the n-th element of the
diagonal. As there are infinitely many n's, there are infinitely many
elements in the diagonal. If there *would* be an infinitieth line there
would be an infinitieth element in the diagonal, which would make it an
invalid representation.
WM
> In article <0df634ab-40c2-47f3-a76e-a956ef011...@h60g2000hsg.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 6 Okt., 14:34, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> ...
> > > > If the anti-diagonal, has aleph_0 digits, then there is at least one
> > > > list entry with aleph_0 - 1 digits, by construction. How much is that?
> > >
> > > Not so. I see no construction there.
> >
> > 1
> > 11
> > 111
> > ...
> >
> > You can't see that there is no diagonal with X elements unless there
> > is at least one line with X elements? Sorry, then we are discussing in
> > different worlds.
>
> Apparently we are. I am talking mathematics,
No, you are talking superstition.
> what are you talking? Anyway
> in your list above I still do not see a construction.
Perhaps that will help?
1
11
111
1111
11111
...
Or should I extend the figure?
Or should I say that for every digit 1 at the n-th position of the
diagonal there is a line with n 1's ending at the same position?
> What I see that each
> n-th line has an n-th element which provides the n-th element of the
> diagonal. As there are infinitely many n's, there are infinitely many
> elements in the diagonal. If there *would* be an infinitieth line there
> would be an infinitieth element in the diagonal, which would make it an
> invalid representation.
If there are n digits existing in the diagonal, then there is an n-th
digit existing. This holds for every existing digit of the diagonal. I
do not want to talk about non-existing digits.
Regards, WM
Virgil
<6b7355ac-3db4-45ba...@m3g2000hsc.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
If no line ends then the anti-diagonal need not either, so your argument
confounds and contradicts itself.
>
> > At least in any mathematical world which allows a set of all naturals
>
> The last sentence contains a self-contradiction.
The last "sentence" is not a even sentence.
Much less a self-contradictory one.
Virgil
<9ecae13e-4ab8-4537...@y79g2000hsa.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 7 Okt., 14:55, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > In article
> > <0df634ab-40c2-47f3-a76e-a956ef011...@h60g2000hsg.googlegroups.com> WM
> > <mueck...@rz.fh-augsburg.de> writes:
> > > On 6 Okt., 14:34, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > ...
> > > > > If the anti-diagonal, has aleph_0 digits, then there is at least
> > one
> > > > > list entry with aleph_0 - 1 digits, by construction. How much is
> > that?
> > > >
> > > > Not so. I see no construction there.
> > >
> > > 1
> > > 11
> > > 111
> > > ...
> > >
> > > You can't see that there is no diagonal with X elements unless there
> > > is at least one line with X elements? Sorry, then we are discussing in
> > > different worlds.
> >
> > Apparently we are. I am talking mathematics,
>
> No, you are talking superstition.
It is WM who is full of superstitions, such as that ideas without
physical instances cannot exist.
>
> > what are you talking? Anyway
> > in your list above I still do not see a construction.
>
> Perhaps that will help?
>
> 1
> 11
> 111
> 1111
> 11111
> ...
>
> Or should I extend the figure?
Yes. And don't come back until you finish it.
>
> Or should I say that for every digit 1 at the n-th position of the
> diagonal there is a line with n 1's ending at the same position?
Would there be any point to saying that? One sees that there could be
such a line, but a list can easily be formed without any such line.
>
> > What I see that each
> > n-th line has an n-th element which provides the n-th element of the
> > diagonal. As there are infinitely many n's, there are infinitely many
> > elements in the diagonal. If there *would* be an infinitieth line there
> > would be an infinitieth element in the diagonal, which would make it an
> > invalid representation.
>
> If there are n digits existing in the diagonal, then there is an n-th
> digit existing. This holds for every existing digit of the diagonal. I
> do not want to talk about non-existing digits.
Then stop doing so.
No one is forcing you to do all the nonsensical things you keep doing.
>
> Regards, WM
WM
> According to WM's original statement
> "For all X, there is no diagonal with X elements
> unless there is at least one line with X elements"
> X is the cardinality of the number of characters in the anti-diagonal,
> but the index set, N, has a cardinality greater than any of its members,
> so that WM is wrong once again.
That means the diagonal is longer than any of the lines establishing
it. A self-supporting cantilever like that?
x
xx
xxx
xxxx
x
x
May be you can succeed in architecture or civil engineering with that
idea. Not in mathematics!
Regards, WM
Virgil
<957381e6-4762-4433...@a1g2000hsb.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 7 Okt., 07:26, Virgil <Vir...@gmale.com> wrote:
>
>
> > According to WM's original statement
> > "For all X, there is no diagonal with X elements
> > unless there is at least one line with X elements"
> > X is the cardinality of the number of characters in the anti-diagonal,
> > but the index set, N, has a cardinality greater than any of its members,
> > so that WM is wrong once again.
>
> That means the diagonal is longer than any of the lines establishing
> it.
It does not mean that to me. it allows that, but does not rquire that.
A self-supporting cantilever like that?
>
> x
> xx
> xxx
> xxxx
> x
> x
>
> May be you can succeed in architecture or civil engineering with that
> idea. Not in mathematics!
On the contrary, it is perfectly feasible in mathematics but not in
either architecture or civil engineering.
WM, as usual, has things bass ackwards.
>
> Regards, WM
WM
>
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 7 Okt., 07:26, Virgil <Vir...@gmale.com> wrote:
>
> > > According to WM's original statement
> > > "For all X, there is no diagonal with X elements
> > > unless there is at least one line with X elements"
> > > X is the cardinality of the number of characters in the anti-diagonal,
> > > but the index set, N, has a cardinality greater than any of its members,
> > > so that WM is wrong once again.
>
> > That means the diagonal is longer than any of the lines establishing
> > it.
>
No? Then there could be a line establishing the diagonal and being as
long as the diagonal?
>
> > A self-supporting cantilever like that?
>
>
>
> > x
> > xx
> > xxx
> > xxxx
> > x
> > x
>
> > May be you can succeed in architecture or civil engineering with that
> > idea. Not in mathematics!
>
> On the contrary, it is perfectly feasible in mathematics
No.
Regards, WM
Virgil
<74d731bd-69fb-45d9...@a19g2000pra.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 7 Okt., 21:22, Virgil <Vir...@gmale.com> wrote:
> > In article
> > <957381e6-4762-4433-b9c4-4b961ab65...@a1g2000hsb.googlegroups.com>,
> >
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 7 Okt., 07:26, Virgil <Vir...@gmale.com> wrote:
> >
> > > > According to WM's original statement "For all X, there is no
> > > > diagonal with X elements unless there is at least one line
> > > > with X elements" X is the cardinality of the number of
> > > > characters in the anti-diagonal, but the index set, N, has a
> > > > cardinality greater than any of its members, so that WM is
> > > > wrong once again.
> >
> > > That means the diagonal is longer than any of the lines
> > > establishing it.
> >
> > It does not mean that to me. it allows that, but does not require
> > that.
>
> No? Then there could be a line establishing the diagonal and being as
> long as the diagonal?
There could be all lines in the list as long as the diagonal, namely
endless.
> >
> > > A self-supporting cantilever like that?
> >
> >
> >
> > > x xx xxx xxxx x x
> >
> > > May be you can succeed in architecture or civil engineering with
> > > that idea. Not in mathematics!
> >
> > On the contrary, it is perfectly feasible in mathematics but, not in
> > either architecture or civil engineering.
>
> No.
If WM thinks "no" then he does not understand mathematics as well as he
thinks he does.
>
> Regards, WM
David R Tribble
>> According to WM's original statement
>> "For all X, there is no diagonal with X elements
>> unless there is at least one line with X elements"
>> X is the cardinality of the number of characters in the anti-diagonal,
>> but the index set, N, has a cardinality greater than any of its members,
>> so that WM is wrong once again.
>
WM wrote:
> That means the diagonal is longer than any of the lines establishing
> it. A self-supporting cantilever like that?
> x
> xx
> xxx
> xxxx
> ...
Your own diagram shows that this is true.
At each line n, there is an nth digit of the diagonal. And at each
line n, according to all of the diagrams you have drawn, there
is also a next line n+1. Therefore there is always a next n+1
digit of the diagonal. Therefore the diagonal always has at least
one more digit than the digits accumulated up to line n.
Therefore the diagonal has more digits than any line in your list.
Which means that "the diagonal is longer than any of the lines
establishing it".
If you don't agree with the conclusion drawn from your own
diagram, perhaps there is something wrong with the way you
drew your list?
Dik T. Winter
> On 7 Okt., 14:55, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
...
> > > > > If the anti-diagonal, has aleph_0 digits, then there is at least
> > > > > one list entry with aleph_0 - 1 digits, by construction. How much
> > > > > is that?
> > > >
> > > > Not so. I see no construction there.
> > >
> > > 1
> > > 11
> > > 111
> > > ...
> > >
> > > You can't see that there is no diagonal with X elements unless there
> > > is at least one line with X elements? Sorry, then we are discussing in
> > > different worlds.
> >
> > Apparently we are. I am talking mathematics,
>
> No, you are talking superstition.
That is *your* opinion.
> > what are you talking? Anyway
> > in your list above I still do not see a construction.
>
> Perhaps that will help?
>
> 1
> 11
> 111
> 1111
> 11111
> ...
Still no construction.
> Or should I extend the figure?
A figure is not a construction.
> Or should I say that for every digit 1 at the n-th position of the
> diagonal there is a line with n 1's ending at the same position?
But that is right. And in the complete diagonal there is for every 1 of
the diagonal a finite line ending with an 1 in that position. And there
are infinitely many such positions.
> > What I see that each
> > n-th line has an n-th element which provides the n-th element of the
> > diagonal. As there are infinitely many n's, there are infinitely many
> > elements in the diagonal. If there *would* be an infinitieth line there
> > would be an infinitieth element in the diagonal, which would make it an
> > invalid representation.
>
> If there are n digits existing in the diagonal, then there is an n-th
> digit existing. This holds for every existing digit of the diagonal. I
> do not want to talk about non-existing digits.
This holds *only* for n finite. When n is aleph_0, there is *no* aleph_0-th
digit. That is where your thinking goes wrong. When n is aleph_0 that means
only that there is no *last* digit.