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Topic: Continuum Hypothesis Solution Posted
Replies: 44   Last Post: May 1, 1999 5:30 PM

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Nathaniel Deeth

Posts: 548
Registered: 12/4/04
Re: Continuum Hypothesis Solution Posted
Posted: Apr 21, 1999 5:30 AM
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Ulrich Weigand wrote:

> Nathan the Great <mad@ashland.baysat.net> writes:
>

> > x = 1 ;the first number
> > N = x ;Set N = {1}
> > Do
> > x = x + 1 ;the successor of x
> > N = N U x ;union element x to Set N
> > Loop

>
> >If/when this algorithm *completes*, N will contain all the natural numbers.
> >And according to Cantorians, this never-ending algorithm can, in the Platonic Realm,
> >be ended.

>
> Nope. This algorithm does not terminate. Nevertheless, the set of all
> natural numbers exists. (Who ever said that whether a set does or does
> not exist depends on the existence of an *algorithm* that produces this
> set? Every set produced by an algorithm in this sense will always be
> finite, of course.)
>
> --
> Ulrich Weigand,
> IMMD 1, Universitaet Erlangen-Nuernberg,
> Martensstr. 3, D-91058 Erlangen, Phone: +49 9131 85-7688


Mr. Ulrich, I have two questions:

(1) Is Cantor's Diagonal Number a completed infinity?
(2) Does the "one-fell-swoop" in the proof below terminate?

An irrational number can be described as an infinite string of decimal digits, or, an
infinite series of nested rational intervals. These two descriptions are combined
(below) to repressent pi.

Left: {3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...}
Right: {4, 3.2, 3.15, 3.142, 3.1416, 3.14160, ...}

In order to prove pi is rational, {3, 3.1, 3.14, ...} will be diagonalized.

Instead of constructing a diagonal number which differs, from every listed number, in its

diagonal digit location, Nathan will construct a diagonal number that is similar, to each

listed number, at that location. Clearly, using this method results in Nathan's Diagonal

Number (NDN) being equal to a rational number from the list at each step in its
construction.

row 1.23456 Diagonal
--- ------- --------
1 3 3
2 3.1 1
3 3.14 4
4 3.141 1
5 3.1415 5
6 3.14159 9

Now, in one-fell-swoop construct all the digits of NDN. This changes NDN into a
completed infinity.

Conclusion:

First, in the construction of each specific digit of NDN, the list must contain a
rational number that extends to that digit's location. This pre-existing listed rational

number, as far as it extends, is equal to NDN. Hence, NDN is rational (as far as its
digits extend). Second, at every decimal location, the digit of NDN matches the digit of

pi. Hence, NDN = pi. Therefore, since NDN = pi and NDN is rational, pi is rational.


Nathan the Great
Age 11









Date Subject Author
4/14/99
Read Continuum Hypothesis Solution Posted
Papus
4/15/99
Read Re: Continuum Hypothesis Solution Posted
Webster Kehr
4/16/99
Read Re: Continuum Hypothesis Solution Posted
Alan Morgan
4/18/99
Read Re: Continuum Hypothesis Solution Posted
Webster Kehr
4/16/99
Read Re: Continuum Hypothesis Solution Posted
Dave Seaman
4/16/99
Read Re: Continuum Hypothesis Solution Posted
Bill Taylor
4/16/99
Read Re: Continuum Hypothesis Solution Posted
Nathaniel Deeth
4/16/99
Read Re: Continuum Hypothesis Solution Posted
Jake Wildstrom
4/19/99
Read Re: Continuum Hypothesis Solution Posted
Sami Aario
4/16/99
Read Re: Continuum Hypothesis Solution Posted
Ken Cox
4/19/99
Read Re: Continuum Hypothesis Solution Posted
Michel Hack
4/20/99
Read Re: Continuum Hypothesis Solution Posted
Nathaniel Deeth
4/20/99
Read Re: Continuum Hypothesis Solution Posted
Ulrich Weigand
4/21/99
Read Re: Continuum Hypothesis Solution Posted
Nathaniel Deeth
4/21/99
Read Re: Continuum Hypothesis Solution Posted
Nathaniel Deeth
4/21/99
Read Re: Continuum Hypothesis Solution Posted
Ulrich Weigand
4/21/99
Read Re: Continuum Hypothesis Solution Posted
Brian David Rothbach
4/21/99
Read Re: Continuum Hypothesis Solution Posted
Virgil Hancher
4/22/99
Read Re: Continuum Hypothesis Solution Posted
Nathaniel Deeth
4/22/99
Read Re: Continuum Hypothesis Solution Posted
Sami Aario
4/23/99
Read Re: Continuum Hypothesis Solution Posted
Nathaniel Deeth
4/25/99
Read Re: Continuum Hypothesis Solution Posted
Sami Aario
4/22/99
Read Re: Continuum Hypothesis Solution Posted
Ulrich Weigand
4/23/99
Read Re: Continuum Hypothesis Solution Posted
Nathaniel Deeth
4/23/99
Read Re: Continuum Hypothesis Solution Posted
Ulrich Weigand
4/20/99
Read Re: Continuum Hypothesis Solution Posted
Nathaniel Deeth
4/17/99
Read Re: Bill Taylor's comments on uncomputable reals
Papus
4/19/99
Read Re: Bill Taylor's comments on uncomputable reals
Kevin Lacker
4/20/99
Read Re: Bill Taylor's comments on uncomputable reals
Bill Taylor
4/19/99
Read Re: Bill Taylor's comments on uncomputable reals
Andrew Boucher
4/19/99
Read Re: Bill Taylor's comments on uncomputable reals
Kevin Lacker
4/21/99
Read Re: Bill Taylor's comments on uncomputable reals
Andrew Boucher
4/22/99
Read Re: Bill Taylor's comments on uncomputable reals
Keith Ramsay
4/23/99
Read Re: Bill Taylor's comments on uncomputable reals
Andrew Boucher
4/24/99
Read Re: Bill Taylor's comments on uncomputable reals
Keith Ramsay
4/25/99
Read Re: Bill Taylor's comments on uncomputable reals
Andrew Boucher
4/27/99
Read Re: Bill Taylor's comments on uncomputable reals
Bill Taylor
4/27/99
Read Re: Bill Taylor's comments on uncomputable reals
David Petry
4/30/99
Read Re: Bill Taylor's comments on uncomputable reals
Keith Ramsay
5/1/99
Read Re: Bill Taylor's comments on uncomputable reals
Keith Ramsay
4/16/99
Read Re: Continuum Hypothesis Solution Posted
Andrew Boucher
4/16/99
Read Re: Continuum Hypothesis Solution Posted
Dave Seaman
4/18/99
Read Re: Continuum Hypothesis Solution Posted
Webster Kehr
4/19/99
Read Re: Continuum Hypothesis Solution Posted
Jeremy Boden
4/19/99
Read Re: Continuum Hypothesis Solution Posted
Sami Aario

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