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Christian Stapfer
science forum beginner

Joined: 05 Jun 2006
Posts: 15

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

Abstract Dissonance wrote:

 Quote: Christian Stapfer wrote:

You did not answer even a single sentence that

Regards,
Christian
Mike Kelly
science forum Guru Wannabe

Joined: 30 Mar 2006
Posts: 119

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

Abstract Dissonance wrote:
 Quote: "Christian Stapfer" wrote in message news:6bd7f\$4483d19c\$54482e3f\$25402@news.hispeed.ch... "Mike Kelly" 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. .
Ben Rudiak-Gould
science forum Guru

Joined: 04 May 2005
Posts: 382

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

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

-- Ben
Russell Easterly
science forum Guru Wannabe

Joined: 27 Jun 2005
Posts: 199

 Posted: Mon Jun 05, 2006 6:58 pm    Post subject: Re: The list of all natural numbers don't exist "Albrecht" 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
guenther.vonKnakspott@gmx
science forum Guru Wannabe

Joined: 24 Apr 2005
Posts: 250

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

Russell Easterly wrote:
 Quote: "Albrecht" 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...
albstorz@gmx.de
science forum Guru Wannabe

Joined: 11 Sep 2005
Posts: 241

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

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
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 ) 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.
Peter Niessen
science forum beginner

Joined: 05 Jun 2006
Posts: 19

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

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
news:de.alt.gruppenkasper ??
Gruppenkasper is Trademark of German Fools
--
Mit freundlichen Grüssen
Peter Nießen
David R Tribble
science forum Guru

Joined: 21 Jul 2005
Posts: 1005

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

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).
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

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

"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.
Peter Webb
science forum Guru Wannabe

Joined: 05 May 2005
Posts: 192

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

"Albrecht" <albstorz@gmx.de> wrote in message
 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.
Dik T. Winter
science forum Guru

Joined: 25 Mar 2005
Posts: 1359

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

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
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/
Ross A. Finlayson
science forum Guru

Joined: 30 Apr 2005
Posts: 873

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

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
Gene Ward Smith
science forum Guru

Joined: 08 Jul 2005
Posts: 409

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

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.
guenther.vonKnakspott@gmx
science forum Guru Wannabe

Joined: 24 Apr 2005
Posts: 250

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

Dik T. Winter wrote:
 Quote: In article <1149541177.093533.90850@f6g2000cwb.googlegroups.com> "Albrecht" 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.
Han de Bruijn
science forum Guru

Joined: 18 May 2005
Posts: 1285

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

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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