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The list of all natural numbers don't exist
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Christian Stapfer
science forum beginner


Joined: 05 Jun 2006
Posts: 15

PostPosted: Mon Jun 05, 2006 11:41 am    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Abstract Dissonance wrote:

Quote:
Christian Stapfer wrote:

You did not answer even a single sentence that
I had written myself: so please quote correctly.

Regards,
Christian
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Mike Kelly
science forum Guru Wannabe


Joined: 30 Mar 2006
Posts: 119

PostPosted: Mon Jun 05, 2006 12:00 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Abstract Dissonance wrote:
Quote:
"Christian Stapfer" <nil@dev.nul> wrote in message
news:6bd7f$4483d19c$54482e3f$25402@news.hispeed.ch...
"Mike Kelly" <mk4284@bris.ac.uk> wrote in message
news:1149457274.133285.140760@y43g2000cwc.googlegroups.com...

Albrecht wrote:
The list of all natural numbers don't exist


Assumption: The list of all natural numbers exists.

The following sketch shows the sequence of the natural numbers in a
mono-cipher representation:

X
XX
XXX
XXXX
.
.
.


Since any next line contains the next natural number starting from 1
("X"), every line under the first line contains a natural number.
Since there are infinite many natural numbers, there are infinite many
"X" in the first column.

The next sketch shows the sequence of natural numbers in which the
digits are numbered with their index numbers. The index numbers are in
mono-cipher representation too, written with the cipher "0" and in
vertical orientation.

0000 . . .
X000
XX00
XXX0
XXXX
.
.
.

The sequence of the vertical sets of "0" represents the sequence of
the natural numbers as the sequence of the horizontal sets of the
"X" represents the natural numbers. So, in both sequences hold the
same sentence: since no natural number contains infinite many digits no
horizontal set of "X" and no vertical set of "0" is infinite in
size.

In consequence, there must be more "X" in the first column as there
are "0" in any column and there must be more "0" in the first
line as there are "X" in any line.

True.


False. Are there more odd positive numbers than even? After all, you start
with 1 so surely there is atleast one more odd?

There are countably infinite many positive odd integers and countably
infinite many even odd integers. There are the same amount of each. I
don't see the relevence to the OP statement that there are more "X" in
the first column than there are "0" in any column. He doesn't claim
there are more "X" in the first column than there are "0" in the first
row.

There are countably infinite many "X" in the first column and a finite
number of "0" in any column. It is correct to say there are more "X" in
the first column than there are "0" in any column.

Similarly there are countably infinite many "0" in the first row and a
finite number of "X" in any row. It is correct to say there are more
"0" in the first column than there are "X" in any row.


Quote:
The fact there are the same amount... both are infinite. Notice also that
nth column in a list with n columns has exactly the same 0's as the X's in
the first column.
.
.
Note that the "process" of ... and . should be shown to be commutable. You
cannot treat the second as a finite process and then the first as an
infinite one or vice versa.

Proof? Why isn't

XXXX....
0XXX
00XX
000X
0000
.
.
.

Just as relvant? i.e., (swaping the X's and O's results in an isomorphic
representation).

i.e., by the original posters same argument there are more 0's than X's.
(hence they must be the same)

Jon

I don't agree with the OP argument. The first error is in the statement
"there must be at least two lines in sequence, which contains (at least
one) "X" but no "0" and there must be at least one column, which
contains (at least one) "0" but no "X". "

--
mike. .
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Ben Rudiak-Gould
science forum Guru


Joined: 04 May 2005
Posts: 382

PostPosted: Mon Jun 05, 2006 3:19 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Albrecht wrote:
Quote:
The list of all natural numbers don't exist

I wacky-parsed that as "the list of all natural numbers that don't exist",
which I thought was an intriguing concept. I figured it would be some silly
paradox like the one about the smallest uninteresting number.

-- Ben
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Russell Easterly
science forum Guru Wannabe


Joined: 27 Jun 2005
Posts: 199

PostPosted: Mon Jun 05, 2006 6:58 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

"Albrecht" <albstorz@gmx.de> wrote in message
news:1149454615.256291.79900@u72g2000cwu.googlegroups.com...
The list of all natural numbers don't exist

You might be interested in an old thread called:
No Set Contains Every Computable Natural.

That thread assumed there exists a tape with
every natural number encoded as a string of 1's
followed by a blank space.

There is a simple Turing Machine that shows
this tape does not contain every natural number.


Russell
- 2 many 2 count
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guenther.vonKnakspott@gmx
science forum Guru Wannabe


Joined: 24 Apr 2005
Posts: 250

PostPosted: Mon Jun 05, 2006 7:23 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Russell Easterly wrote:
Quote:
"Albrecht" <albstorz@gmx.de> wrote in message
news:1149454615.256291.79900@u72g2000cwu.googlegroups.com...
The list of all natural numbers don't exist

You might be interested in an old thread called:
No Set Contains Every Computable Natural.

That thread assumed there exists a tape with
every natural number encoded as a string of 1's
followed by a blank space.

There is a simple Turing Machine that shows
this tape does not contain every natural number.


Russell
- 2 many 2 count
It is amazing how many morons can come up with what they deem to be

smartly constructed lists of the naturals and methods for proving they
are incomplete... And they also never fail to deduce from their
deplorable crap that every list of the naturals is incomplete...
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albstorz@gmx.de
science forum Guru Wannabe


Joined: 11 Sep 2005
Posts: 241

PostPosted: Mon Jun 05, 2006 8:59 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Reef Fish wrote:
Quote:
Albrecht wrote:
The list of all natural numbers don't exist

Of course the list exists -

Can you show me the proof that this list exists? I've never seen a
proof like this. I only have seen a lots of statements which state that
this list had to exist.
Please don't mingle the _actual list_ or the _actual sequence_ of all
natural numbers with the normal mathematical understanding of the
sequence of the natural numbers.
I don't disagree with the following sentences:
- there are infinite many natural numbers
- the sequence of the natural numbers exists
but I disagree with the following sentences:
- the (actual) set of the natural numbers exist
- the actual sequence or actual list of all natural numbers exists

The fine difference perhaps is better understandable when you
understand the sentence which goes back to Georg Cantor:
"The list of all real numbers don't exist."
My proof disproof the proof of Georg Cantor (the proof is called the
diagonal argument or second diagonal proof of Georg Cantor).

Best regards
Albrecht S. Storz


Quote:
- that's how you tell denumerable infinity
from nondenumerable (or uncountable) infinity.

It exists, but is COUNTABLY infinite.

As a matter of fact, there is a difference between an EFFECTIVELY
countable sequence, and one that is not effectively countable,
but countably infinite nevertheless.

All this came back from the Foundations of Mathematics courses
I had taken (three different courses in the Foundations, all
perfectly useless for me today <G>) from text books written by
such Foundations gurus as Sierpinski, Kolmogorov, and other
Russian and Polish mathematicians.

Assumption: The list of all natural numbers exists.

That's the consequence of the Axiomatic development of the
Real Number System, and the development of the cardinal
numbers.

-- Bob.
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Peter Niessen
science forum beginner


Joined: 05 Jun 2006
Posts: 19

PostPosted: Mon Jun 05, 2006 9:25 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Am 5 Jun 2006 13:59:37 -0700 schrieb Albrecht:

Full Quoting and X-Posting for medical diagnostic:

Quote:
Can you show me the proof that this list exists? I've never seen a
proof like this. I only have seen a lots of statements which state that
this list had to exist.
Please don't mingle the _actual list_ or the _actual sequence_ of all
natural numbers with the normal mathematical understanding of the
sequence of the natural numbers.
I don't disagree with the following sentences:
- there are infinite many natural numbers
- the sequence of the natural numbers exists
but I disagree with the following sentences:
- the (actual) set of the natural numbers exist
- the actual sequence or actual list of all natural numbers exists

The fine difference perhaps is better understandable when you
understand the sentence which goes back to Georg Cantor:
"The list of all real numbers don't exist."
My proof disproof the proof of Georg Cantor (the proof is called the
diagonal argument or second diagonal proof of Georg Cantor).

Best regards
Albrecht S. Storz

Really!
Great Theater!
Be patient:
You are not the only Fool Smile
What's about Follow up to:
news:de.alt.gruppenkasper ??
Gruppenkasper is Trademark of German Fools Smile
--
Mit freundlichen Grüssen
Peter Nießen
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David R Tribble
science forum Guru


Joined: 21 Jul 2005
Posts: 1005

PostPosted: Mon Jun 05, 2006 9:34 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Albrecht wrote:
Quote:
The list of all natural numbers don't exist


Jan Burse wrote:
Quote:
Actually the list ends at 666.

Which means the list will end tomorrow (6/6/06).
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Mon Jun 05, 2006 10:56 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

In article <1149508837.399870.70940@f6g2000cwb.googlegroups.com>,
"Mike Kelly" <mk4284@bris.ac.uk> wrote:

Quote:
There are countably infinite many positive odd integers and countably
infinite many even odd integers. There are the same amount of each.

I can see that an even odd integer would be very odd indeed, but I do
not see that there are any of them, much less infinitely many of them.
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Peter Webb
science forum Guru Wannabe


Joined: 05 May 2005
Posts: 192

PostPosted: Tue Jun 06, 2006 12:26 am    Post subject: Re: The list of all natural numbers don't exist Reply with quote

"Albrecht" <albstorz@gmx.de> wrote in message
news:1149541177.093533.90850@f6g2000cwb.googlegroups.com...
Quote:
Reef Fish wrote:
Albrecht wrote:
The list of all natural numbers don't exist

Of course the list exists -

Can you show me the proof that this list exists? I've never seen a
proof like this. I only have seen a lots of statements which state that
this list had to exist.
Please don't mingle the _actual list_ or the _actual sequence_ of all
natural numbers with the normal mathematical understanding of the
sequence of the natural numbers.
I don't disagree with the following sentences:
- there are infinite many natural numbers
- the sequence of the natural numbers exists
but I disagree with the following sentences:
- the (actual) set of the natural numbers exist
- the actual sequence or actual list of all natural numbers exists


I think I can settle this.

Here is my proposed set of all Natural numbers:

S = {1, 2, 3, 4, ....}.

Clearly every member of the set is a Natural number.

I will now prove that it contains all Natural numbers by using the 5th Peano
Axiom (induction).

Clearly 1 is an element of S.

If a natural number k is an element of S, so is k+1 (by construction of S).

So k is an element of S for all natural numbers k.

Therefore all Natural numbers are elements of S.

Therefore S is exactly the set of all Natural numbers.
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Dik T. Winter
science forum Guru


Joined: 25 Mar 2005
Posts: 1359

PostPosted: Tue Jun 06, 2006 1:02 am    Post subject: Re: The list of all natural numbers don't exist Reply with quote

In article <1149541177.093533.90850@f6g2000cwb.googlegroups.com> "Albrecht" <albstorz@gmx.de> writes:
....
Quote:
Can you show me the proof that this list exists? I've never seen a
proof like this. I only have seen a lots of statements which state that
this list had to exist.

Yes, it does.

Quote:
Please don't mingle the _actual list_ or the _actual sequence_ of all
natural numbers with the normal mathematical understanding of the
sequence of the natural numbers.
I don't disagree with the following sentences:
- there are infinite many natural numbers
- the sequence of the natural numbers exists

And that is a list as it gives a one-to-one mapping from the natural
numbers to the elements in the sequence. Do you know the meaning of
the mathematical term "list"? More informal, you give a list when
within finite time it can be determined what the n'th element of the
list is for each natural n.

Quote:
but I disagree with the following sentences:
- the (actual) set of the natural numbers exist
- the actual sequence or actual list of all natural numbers exists

Not knowing what you mean with "actual list", I have no idea.

Quote:
The fine difference perhaps is better understandable when you
understand the sentence which goes back to Georg Cantor:
"The list of all real numbers don't exist."

Which means: there is no one-to-one mapping from the natural numbers
to the real numbers. (Note that the paraphrase is not talking about
"actual list", whatever that may mean, the use of the term "list" is
done in the mathematical sense.)

Quote:
My proof disproof the proof of Georg Cantor

Your proof fails, because when you apply the diagonal argument to the
natural numbers you obviously get something that is not on the list.
(And I am talking about "list", not about "actual list", whatever that
may mean.) But you have not shown that what you obtain is indeed a
natural number. On the other hand, the number you get by the diagonal
argument with a list of real numbers is a real number by the definition
of real numbers. Now try to apply the diagonal argument to rational
numbers written in decimal (probably non-terminating) form. You get a
new number, but that does *not* show that the list is incomplete because
you do not know whether the new number is rational or real. (It gets
a bit murky when you talk about the list of computable numbers. In this
case a list does exist, but the list itself is not computable, so the
diagonal number is not computable.)
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
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Ross A. Finlayson
science forum Guru


Joined: 30 Apr 2005
Posts: 873

PostPosted: Tue Jun 06, 2006 1:25 am    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Aatu Koskensilta wrote:
Quote:
Christian Stapfer wrote:
A word of caution: You're probably right in claiming
that Albrecht hasn't managed show ZFC to be "self
contradicting". But you would be quite mistaken in
claiming that to say that "ZFC is self contradicting"
is "definitely false". That statement has currently
the same status as its opposite, namely that "ZFC
is consistent (not self contradicting)": it's truth
value is not *known*. And by *known* I do not mean
'hoped for' (Hilbert?) or 'pragmatically assumed'
(Bourbaki) or 'directly *seen* by way of a Platonist
view of the universe of sets' (Gödel) or 'taken
for granted' or some such thing...

The consistency of ZFC is a trivial consequence of the truth of the
currently accepted principles of set theory, so if one accepts these
principles there is nothing problematic in the consistency of ZFC.
Whether we 'know' these principles can of course be endlessly debated,
but if we don't know them then there is no basis to claim that we know
theorems of ZFC (not known to provable from weaker theories) but don't
know "ZFC is consistent"; these theorems are then equally 'hoped for',
'seen' or 'taken for granted'.

--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus


Yaroslav Sergeyev posits the existence of a unit infinity, similar to
what I call a unit scalar infinity or Tony here a unit infinity, that
he calls a grossone, in his book Arithmetic of Infinity. I wonder what
you think of it. I tell him that the existence of such an object
contradicts some notions held by writers to these lists, he axiomatizes
its presence.

In unary, base one, or base infinity, in the coded representation of
the unit interval of real numbers, the antidiagonal argument doesn't
hold, and neither does nested intervals.

Consider Boucher's recent presentation of reasoning to deny Goedel's
incompleteness.

Consider model theory. There is a fiat that some ordinal element M of
M the model is a universal ordinal, no? There is no universe in ZF(C).
Where, via unrestricted comprehension there is a universe, it is a
true statement that the universe exists. Yet, its existence is denied
in ZF(C). V = L is not a meaningful statement when neither of those
objects, V the universe nor L the constructible universe, exist in ZF
set theory. There is no Goedel or arbitrarily von Neumann or other
ordinal coding of those concepts.

In the null axiom theory, as ordinals: the powerset is order type is
succcessor, and there are ubiquitous naturals.

Ross
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Gene Ward Smith
science forum Guru


Joined: 08 Jul 2005
Posts: 409

PostPosted: Tue Jun 06, 2006 5:25 am    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Ross A. Finlayson wrote:

Quote:
Yaroslav Sergeyev posits the existence of a unit infinity, similar to
what I call a unit scalar infinity or Tony here a unit infinity, that
he calls a grossone, in his book Arithmetic of Infinity. I wonder what
you think of it. I tell him that the existence of such an object
contradicts some notions held by writers to these lists, he axiomatizes
its presence.

That's not a book many people here will have a chance to see, so I
wonder if you can explain what the "grossone" is. It's standard to
introduce a unit infinity into the reals in some sense, since if we
take R(x) or R((1/x)) and make x larger than any real, we can be said
to have done that.
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guenther.vonKnakspott@gmx
science forum Guru Wannabe


Joined: 24 Apr 2005
Posts: 250

PostPosted: Tue Jun 06, 2006 5:29 am    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Dik T. Winter wrote:
Quote:
In article <1149541177.093533.90850@f6g2000cwb.googlegroups.com> "Albrecht" <albstorz@gmx.de> writes:
...
snip
Your proof fails, because when you apply the diagonal argument to the
natural numbers you obviously get something that is not on the list.
(And I am talking about "list", not about "actual list", whatever that
may mean.) But you have not shown that what you obtain is indeed a
natural number. On the other hand, the number you get by the diagonal
argument with a list of real numbers is a real number by the definition
of real numbers. Now try to apply the diagonal argument to rational
numbers written in decimal (probably non-terminating) form. You get a
new number, but that does *not* show that the list is incomplete because
you do not know whether the new number is rational or real. (It gets
a bit murky when you talk about the list of computable numbers. In this
case a list does exist, but the list itself is not computable, so the
diagonal number is not computable.)
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

It doesn't look as if this particular moron where construing a
"diagonal" number. His argument seems to be something like "the entries
in the list are not in a 1-1 corresponcence to their indexes".
They come in all shapes and flavours, you know...
Regards.
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Han de Bruijn
science forum Guru


Joined: 18 May 2005
Posts: 1285

PostPosted: Tue Jun 06, 2006 8:08 am    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Albrecht wrote:

Quote:
The list of all natural numbers don't exist

Yes, everybody knows that it doesn't really exist.

Han de Bruijn
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