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The list of all natural numbers don't exist
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Jan Burse
science forum addict


Joined: 27 Jan 2006
Posts: 85

PostPosted: Sun Jun 04, 2006 9:00 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Actually the list ends at 666.

Albrecht wrote:

Quote:
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.

Now we can conclude from the second sketch that 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".
Else there are not infinitely many natural numbers or there are natural
numbers with infinitely many digits - witch is both impossible.

If there are two lines in sequence, let's call them o and p, which
contains "X" but no "0", there is a following line q containing
a natural number which do not have an index number for every digit.
This is impossible since every digit of every natural number must have
an index number.

If there is a column, which contains "0" but no "X", the next
column contains an index number, which don't correspond to a digit of
a natural number. This is impossible since for every index number N
there exists at least one natural number with N digits.

In consequence, the list of all natural numbers can't exist.

The shown contradictions have a great impact on axiomatic set theory:
- The proof Cantor's about the existence of more real numbers than
natural numbers don't hold
- The existence of infinite sets is self contradicting
- ZFC is self contradicting


Best regards

Albrecht Siegfried Storz
Mannheim, Germany


This is the translation of the first posting of the thread "Die Liste
aller natürlichen Zahlen existiert NICHT"
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Reef Fish
science forum Guru Wannabe


Joined: 28 Apr 2005
Posts: 200

PostPosted: Sun Jun 04, 2006 9:14 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

Of course the list exists -- 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.

Quote:
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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Mike Kelly
science forum Guru Wannabe


Joined: 30 Mar 2006
Posts: 119

PostPosted: Sun Jun 04, 2006 9:41 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


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.

Quote:
Now we can conclude from the second sketch that 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".

False. This doesn't follow.

Quote:
Else there are not infinitely many natural numbers or there are natural
numbers with infinitely many digits - witch is both impossible.

This doesn't follow.

Quote:
If there are two lines in sequence, let's call them o and p, which
contains "X" but no "0",

There isn't. Every row contains a countably infinite number of "0" and
a finite number of "X".

Quote:
there is a following line q containing
a natural number which do not have an index number for every digit.
This is impossible since every digit of every natural number must have
an index number.

If there is a column, which contains "0" but no "X",

There isn't. Every column contains a countably infinite number of "X"
and a finite number of "0".

Quote:
the next
column contains an index number, which don't correspond to a digit of
a natural number. This is impossible since for every index number N
there exists at least one natural number with N digits.

In consequence, the list of all natural numbers can't exist.

False.

Quote:
The shown contradictions have a great impact on axiomatic set theory:
- The proof Cantor's about the existence of more real numbers than
natural numbers don't hold

False.

Quote:
- The existence of infinite sets is self contradicting

False.

Quote:
- ZFC is self contradicting

Definately false.

-- mike
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John Jones
science forum beginner


Joined: 08 Feb 2005
Posts: 13

PostPosted: Sun Jun 04, 2006 9:46 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Wow. Isn't it just great that we, the public, can be certain of the
infinite number and its ilk?
Aren't we lucky that we can rely on the experts concerning the
necessary existence of infinite numbers?
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Hero
science forum Guru Wannabe


Joined: 29 Apr 2005
Posts: 220

PostPosted: Sun Jun 04, 2006 10:06 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

John Jones schrieb:

Quote:
Wow. Isn't it just great that we, the public, can be certain of the
infinite number and its ilk?
Aren't we lucky that we can rely on the experts concerning the
necessary existence of infinite numbers?
No, You can stay inside with a finite number. But what if Your daughter

asks:
"If i add one to Your number, what i'll get?"
Hero
PS You can become an expert too, if You want. A few centuries ago
someone who could read and write was considered to be an expert already.
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John Jones
science forum beginner


Joined: 08 Feb 2005
Posts: 13

PostPosted: Sun Jun 04, 2006 10:59 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Hero wrote:
Quote:
But what if Your daughter
asks:
"If i add one to Your number, what i'll get?"

You can't add one to an unnamed number and expect to get a named
number. I hope I am not expected to consider that there are unnamed
numbers, and that these can be used with named numbers.
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Gene Ward Smith
science forum Guru


Joined: 08 Jul 2005
Posts: 409

PostPosted: Sun Jun 04, 2006 11:15 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

Didn't we just have this conversation in German?
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Hero
science forum Guru Wannabe


Joined: 29 Apr 2005
Posts: 220

PostPosted: Sun Jun 04, 2006 11:21 pm    Post subject: Re: The list of all natural numbers don't exist Reply with quote

John Jones schrieb:

Quote:
Hero wrote:
But what if Your daughter
asks:
"If i add one to Your number, what i'll get?"

You can't add one to an unnamed number and expect to get a named
number. I hope I am not expected to consider that there are unnamed
numbers, and that these can be used with named numbers.
You didn't wanted to rely on experts talking about infinite numbers, so

i proposed, You stick to a finite one. You can call it with any name
You like, but if You lay down just as many cents or grains or whatever
You prefer, Your daughter might lay down one more to them.
Just like that.
Hero
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Christian Stapfer
science forum beginner


Joined: 05 Jun 2006
Posts: 15

PostPosted: Mon Jun 05, 2006 6:39 am    Post subject: Re: The list of all natural numbers don't exist Reply with quote

"Mike Kelly" <mk4284@bris.ac.uk> wrote in message
news:1149457274.133285.140760@y43g2000cwc.googlegroups.com...
Quote:

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.

Now we can conclude from the second sketch that 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".

False. This doesn't follow.

Else there are not infinitely many natural numbers or there are natural
numbers with infinitely many digits - witch is both impossible.

This doesn't follow.

If there are two lines in sequence, let's call them o and p, which
contains "X" but no "0",

There isn't. Every row contains a countably infinite number of "0" and
a finite number of "X".

there is a following line q containing
a natural number which do not have an index number for every digit.
This is impossible since every digit of every natural number must have
an index number.

If there is a column, which contains "0" but no "X",

There isn't. Every column contains a countably infinite number of "X"
and a finite number of "0".

the next
column contains an index number, which don't correspond to a digit of
a natural number. This is impossible since for every index number N
there exists at least one natural number with N digits.

In consequence, the list of all natural numbers can't exist.

False.

The shown contradictions have a great impact on axiomatic set theory:
- The proof Cantor's about the existence of more real numbers than
natural numbers don't hold

False.

- The existence of infinite sets is self contradicting

False.

- ZFC is self contradicting

Definately false.

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...

Regards,
Christian
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guenther.vonKnakspott@gmx
science forum Guru Wannabe


Joined: 24 Apr 2005
Posts: 250

PostPosted: Mon Jun 05, 2006 7:28 am    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Albrecht wrote:
<snip crap>
Quote:

Albrecht Siegfried Storz
Mannheim, Germany


This is the translation of the first posting of the thread "Die Liste
aller natürlichen Zahlen existiert NICHT"
Why don't you keep this crap to the kraut groups?
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John Jones
science forum beginner


Joined: 08 Feb 2005
Posts: 13

PostPosted: Mon Jun 05, 2006 8:56 am    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Hero wrote:
Quote:
John Jones schrieb:

Hero wrote:
But what if Your daughter
asks:
"If i add one to Your number, what i'll get?"

You can't add one to an unnamed number and expect to get a named
number. I hope I am not expected to consider that there are unnamed
numbers, and that these can be used with named numbers.
You didn't wanted to rely on experts talking about infinite numbers, so
i proposed, You stick to a finite one. You can call it with any name
You like, but if You lay down just as many cents or grains or whatever
You prefer, Your daughter might lay down one more to them.
Just like that.
Hero

You have to 'add one' to get a larger number from a number. If you
don't 'add one', then you don't get a larger number.
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Aatu Koskensilta
science forum Guru Wannabe


Joined: 17 May 2005
Posts: 277

PostPosted: Mon Jun 05, 2006 10:45 am    Post subject: Re: The list of all natural numbers don't exist Reply with quote

Christian Stapfer wrote:
Quote:
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
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Abstract Dissonance
science forum Guru Wannabe


Joined: 29 Dec 2005
Posts: 201

PostPosted: Mon Jun 05, 2006 10:54 am    Post subject: Re: The list of all natural numbers don't exist Reply with quote

"Christian Stapfer" <nil@dev.nul> wrote in message
news:6bd7f$4483d19c$54482e3f$25402@news.hispeed.ch...
Quote:
"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?

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

.
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Christian Stapfer
science forum beginner


Joined: 05 Jun 2006
Posts: 15

PostPosted: Mon Jun 05, 2006 11:38 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)": its 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

It is not even a consequence: for most mathematicians
it is nothing but what they, failing to have a direct
access to Gödel's universe of sets, take to be "the
currently accepted principles of set theory" themselves.
But consistency and current acceptance are not exactly
the same...

Quote:
of the truth of the currently accepted principles
of set theory, so if one accepts these principles

Yes, but what a big IF indeed! So you are not
arguing anything here: you are just telling me
that you are following the herd....

Quote:
there is nothing problematic in the consistency of ZFC.

Talk of uttering trivialities: if one believes
in ZFC one believes in ZFC. That much is obvious.

Quote:
Whether we 'know' these principles can of course be
endlessly debated,

There still remains the possibility of the derivability
of a direct contradiction, you know: and such a thing
might come to be known quite precisely...

Quote:
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'.

To argue like that is pure nonsense: there is no
way to rationaly justify the use of ZFC by *such*
circular rhetoric alone. Not *everything* is exactly
as problematic as the principles stated by ZFC.

Regards,
Christian
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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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