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albstorz@gmx.de
science forum Guru Wannabe

Joined: 11 Sep 2005
Posts: 241

Posted: Tue Jun 06, 2006 10:20 am    Post subject: Re: The list of all natural numbers don't exist

Han de Bruijn wrote:
 Quote: Albrecht wrote: The list of all natural numbers don't exist Yes, everybody knows that it doesn't really exist.

That is not the point. The point is, that the idea of the actual list
of all natural numbers is not thinkable without logical contradictions
as I try to show.
But the idea of the actual list of all natural numbers is an implicit
assumption of Cantor's diagonal argument. If my proof is correct,
Cantor's proof is disproofed.

Best regards
Albrecht S. Storz
Jesse F. Hughes
science forum Guru

Joined: 24 Mar 2005
Posts: 801

Posted: Tue Jun 06, 2006 11:01 am    Post subject: Re: The list of all natural numbers don't exist

Han de Bruijn <Han.deBruijn@DTO.TUDelft.NL> writes:

 Quote: Albrecht wrote: The list of all natural numbers don't exist Yes, everybody knows that it doesn't really exist.

I don't even know what "really exists" mean, so I guess I don't know
that the list doesn't really exist.

--
"I'm talking about mathematics--hard, brutal, extreme ... pushing your
mind beyond the limits to understand what no one else can because
they're afraid to risk it all, to lose their freaking worthless minds
in the push to know." --James Harris, for the Nike Derivator
Barb Knox

Joined: 28 Apr 2005
Posts: 64

Posted: Tue Jun 06, 2006 11:10 am    Post subject: Re: The list of all natural numbers don't exist

"Albrecht" <albstorz@gmx.de> wrote:

 Quote: Han de Bruijn wrote: Albrecht wrote: The list of all natural numbers don't exist Yes, everybody knows that it doesn't really exist. That is not the point. The point is, that the idea of the actual list of all natural numbers is not thinkable without logical contradictions as I try to show. But the idea of the actual list of all natural numbers is an implicit assumption of Cantor's diagonal argument. If my proof is correct, Cantor's proof is disproofed.

And if wishes were horses then beggars would ride.

Doesn't it give you the slightest pause that "disproving Cantor" is
canonical crank-fodder (followed closely by "disproving Special
Relativity")?
Dave Seaman
science forum Guru

Joined: 24 Mar 2005
Posts: 527

Posted: Tue Jun 06, 2006 11:45 am    Post subject: Re: The list of all natural numbers don't exist

On 6 Jun 2006 03:20:01 -0700, Albrecht wrote:

 Quote: Han de Bruijn wrote: Albrecht wrote: The list of all natural numbers don't exist Yes, everybody knows that it doesn't really exist.

 Quote: That is not the point. The point is, that the idea of the actual list of all natural numbers is not thinkable without logical contradictions as I try to show. But the idea of the actual list of all natural numbers is an implicit assumption of Cantor's diagonal argument. If my proof is correct, Cantor's proof is disproofed.

How so? Cantor's theorem merely says that |X| < |P(X)| for all X. The
proof shows that if f: X -> P(X) is any mapping, then there exists y in
P(X) that is not in the range of f.

The proof does not even mention "lists".

<http://planetmath.org/encyclopedia/ProofOfCantorsTheorem.html>

--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
albstorz@gmx.de
science forum Guru Wannabe

Joined: 11 Sep 2005
Posts: 241

Posted: Tue Jun 06, 2006 11:46 am    Post subject: Re: The list of all natural numbers don't exist

John Jones wrote:
 Quote: Hero wrote: 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.

Yes, you have found the heart of the proof I had shown
You know: A larger quantity must have at least one more member than a
smaller quantity.
And infinite is truely larger than finite.
That's why there must be a line in my sketch (in this special case
exactly at least two lines in sequence) with "X" but no "0" since the
"X" in the first column are infinite many and the "0" in every column
are only finite many.

Greetings
Albrecht S. Storz
Lee Rudolph
science forum Guru

Joined: 28 Apr 2005
Posts: 566

Posted: Tue Jun 06, 2006 12:15 pm    Post subject: Re: The list of all natural numbers don't exist

Dave Seaman <dseaman@no.such.host> writes:

 Quote: How so? Cantor's theorem merely says that |X| < |P(X)| for all X. The proof shows that if f: X -> P(X) is any mapping, then there exists y in P(X) that is not in the range of f. The proof does not even mention "lists".

"The cardinality of the power set of any set X strictly exceeds the
cardinality of X", said Tom listlessly.

Lee Rudolph
William Hughes
science forum Guru

Joined: 05 May 2005
Posts: 355

Posted: Tue Jun 06, 2006 1:45 pm    Post subject: Re: The list of all natural numbers don't exist

Albrecht wrote:

[snip]

 Quote: That's why there must be a line in my sketch (in this special case exactly at least two lines in sequence) with "X" but no "0" since the "X" in the first column are infinite many and the "0" in every column are only finite many.

Nonsense. Each column has only a finite number of "0" 's.
However, there are an infinite number of columns, so the
number of "0" 's is infinite. So we cannot conlcude that
there is a line with "X' but no "0".

-William Hughes
Dave Seaman
science forum Guru

Joined: 24 Mar 2005
Posts: 527

Posted: Tue Jun 06, 2006 1:46 pm    Post subject: Re: The list of all natural numbers don't exist

On 6 Jun 2006 08:15:37 -0400, Lee Rudolph wrote:
 Quote: Dave Seaman writes: How so? Cantor's theorem merely says that |X| < |P(X)| for all X. The proof shows that if f: X -> P(X) is any mapping, then there exists y in P(X) that is not in the range of f. The proof does not even mention "lists". "The cardinality of the power set of any set X strictly exceeds the cardinality of X", said Tom listlessly.

And Swiftly, too.

 Quote: Lee Rudolph

--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
mensanator@aol.compost
science forum Guru

Joined: 24 Mar 2005
Posts: 826

Posted: Tue Jun 06, 2006 5:11 pm    Post subject: Re: The list of all natural numbers don't exist

Lee Rudolph wrote:
 Quote: Dave Seaman writes: How so? Cantor's theorem merely says that |X| < |P(X)| for all X. The proof shows that if f: X -> P(X) is any mapping, then there exists y in P(X) that is not in the range of f. The proof does not even mention "lists". "The cardinality of the power set of any set X strictly exceeds the cardinality of X", said Tom listlessly.

That was worse (better?) than anything in that math joke thread!

 Quote: Lee Rudolph
Ross A. Finlayson
science forum Guru

Joined: 30 Apr 2005
Posts: 873

Posted: Tue Jun 06, 2006 6:43 pm    Post subject: Re: The list of all natural numbers don't exist

Gene Ward Smith wrote:
 Quote: Ross A. Finlayson wrote: 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.

Hi Gene,

It's a slim volume. Basically the first several chapters introduce
very briefly the notions of cardinality, and then Sergeyev posits some
"point at infinity" with as might be surmised some arithmetical
properties. For example, it's composite, evenly divided by any finite
integer, and then something along the lines of asymptotic density is
symbolizable in that way. The presentation is not more than twenty or
thirty paragraphs or so. Basically the point seems to be that the size
of the naturals is some value with a fixed character in terms of its
use within summations and productions of infinite series etcetera, and
that can be expressed in classical ways.

For example, in a paper "Mathematical foundations for the Infinity
Computer", Yaroslav writes "We conclude this Introduction by
emphasizing that the goal of the paper is not to construct a complete
theory of infinity and discuss such concepts as, for example, 'set of
all sets'. In contrast, the problem of infinity is considered from
positions of applied mathematics and theory and practice of
computations -- fields being among the main scientific interests of the
author. A new viewpoint on infinity and the corresponding mathematical
and computer science tools are introduced in the paper in order to give
possibilities to solve applied problems."

Divide by zero. A general reaction to that might be "division by zero
is undefined." Well, it's not, the operation of division occurs with
dividand zero. While that might be so, the result is not well-defined.
There are considerations along the lines of 0/0 = 1, and 1/0 < 2/0,
etcetera, and sometimes those expressions are true, it varies with the
inputs and extra bookkeeping on the inputs as necessary. Take the
square root of negative one, measure the diagonal of the unit square,
etcetera, imaginary and irrational numbers were once undefined.

"The infinite radix of the new system is introduced as the number of
elements of the set N of natural numbers expressed by the numeral (1)
called _grossone_."

Sergeyev thinks that N supplemented with grossone (a unit infinity) is
not a monoid, that N becomes unclosed under addition, where he
indicates there is some N^hat that appears basically similar to the
hyperintegers. Then, in introduction of some infinitesimals, the
systems becomes similar to the hyperreals, N^hat is closed under
addition, but he doesn't address ((1)). So, his system appears to be a
positional/expansion number system with infinite and infinitesimal

I hope that I have correctly represented his words.

Now, were he to indicate that

Sum dx = 1
(1)

then that would be something. There's perhaps not a huge amount of
novelty in that, considering that's the way infinitesimal analysis has
been done for almost five hundred years already.

There is no set of all cardinals.

Ross
albstorz@gmx.de
science forum Guru Wannabe

Joined: 11 Sep 2005
Posts: 241

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

Dave Seaman wrote:
 Quote: On 6 Jun 2006 03:20:01 -0700, Albrecht wrote: Han de Bruijn wrote: Albrecht wrote: The list of all natural numbers don't exist Yes, everybody knows that it doesn't really exist. That is not the point. The point is, that the idea of the actual list of all natural numbers is not thinkable without logical contradictions as I try to show. But the idea of the actual list of all natural numbers is an implicit assumption of Cantor's diagonal argument. If my proof is correct, Cantor's proof is disproofed. How so? Cantor's theorem merely says that |X| < |P(X)| for all X. The proof shows that if f: X -> P(X) is any mapping, then there exists y in P(X) that is not in the range of f. The proof does not even mention "lists".

Since the (necessarily actual) set of the natural numbers don't exists
(without self contradicting, as I tried to show) the (necessarily
actual) powerset of the natural numbers don't exists either (without

Best regards
Albrecht S. Storz
albstorz@gmx.de
science forum Guru Wannabe

Joined: 11 Sep 2005
Posts: 241

Posted: Tue Jun 06, 2006 8:25 pm    Post subject: Re: The list of all natural numbers don't exist

William Hughes wrote:
 Quote: Albrecht wrote: [snip] That's why there must be a line in my sketch (in this special case exactly at least two lines in sequence) with "X" but no "0" since the "X" in the first column are infinite many and the "0" in every column are only finite many. Nonsense. Each column has only a finite number of "0" 's. However, there are an infinite number of columns, so the number of "0" 's is infinite. So we cannot conlcude that there is a line with "X' but no "0".

Maybe the number of all "0"'s in all columns are infinite. I don't

Do you disagree that an infinite number has at least one more than any
finite number?

If the "X"'s in the first column are infinite many and the "0"'s in any
column are finite many and maybe all sequences of "X"'s and "0"'s
starts at the same line downwards, how could it be that there is no
line with "X" but no "0". In that case, how should the number of "X"'s
in the first column be greater than the number of the "0"'s in any
column (not in all columns)?

Best regards
Albrecht S. Storz
Dave Seaman
science forum Guru

Joined: 24 Mar 2005
Posts: 527

Posted: Tue Jun 06, 2006 8:34 pm    Post subject: Re: The list of all natural numbers don't exist

On 6 Jun 2006 13:08:27 -0700, albstorz@gmx.de wrote:
 Quote: Dave Seaman wrote: On 6 Jun 2006 03:20:01 -0700, Albrecht wrote: Han de Bruijn wrote: Albrecht wrote: The list of all natural numbers don't exist Yes, everybody knows that it doesn't really exist. That is not the point. The point is, that the idea of the actual list of all natural numbers is not thinkable without logical contradictions as I try to show. But the idea of the actual list of all natural numbers is an implicit assumption of Cantor's diagonal argument. If my proof is correct, Cantor's proof is disproofed. How so? Cantor's theorem merely says that |X| < |P(X)| for all X. The proof shows that if f: X -> P(X) is any mapping, then there exists y in P(X) that is not in the range of f. The proof does not even mention "lists".

 Quote: Since the (necessarily actual) set of the natural numbers don't exists (without self contradicting, as I tried to show) the (necessarily actual) powerset of the natural numbers don't exists either (without self contradicting).

Let me know when you are actually able to show (not just try to show)
that N does not exist.

--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Tue Jun 06, 2006 8:36 pm    Post subject: Re: The list of all natural numbers don't exist

"Albrecht" <albstorz@gmx.de> wrote:

 Quote: Han de Bruijn wrote: Albrecht wrote: The list of all natural numbers don't exist Yes, everybody knows that it doesn't really exist. That is not the point. The point is, that the idea of the actual list of all natural numbers is not thinkable without logical contradictions as I try to show. But the idea of the actual list of all natural numbers is an implicit assumption of Cantor's diagonal argument. If my proof is correct, Cantor's proof is disproofed. Best regards Albrecht S. Storz

In systems allowing the Peano postulates as either axioms or theorems, a
"list" of all natural numbers must exist.
John Jones
science forum beginner

Joined: 08 Feb 2005
Posts: 13

Posted: Tue Jun 06, 2006 8:55 pm    Post subject: Re: The list of all natural numbers don't exist

Albrecht wrote:
 Quote: Yes, you have found the heart of the proof I had shown You know: A larger quantity must have at least one more member than a smaller quantity. And infinite is truely larger than finite. That's why there must be a line in my sketch (in this special case exactly at least two lines in sequence) with "X" but no "0" since the "X" in the first column are infinite many and the "0" in every column are only finite many. Greetings Albrecht S. Storz

I still say you have to 'add one' to get a larger number from a number.
Simply 'add one' means nothing. You have to have a number to add to.

It's no good invoking the idea of a 'smaller number' to which one is
'added'. There are no such smaller numbers here UNTIL one is added.

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