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

 Messages: [ Previous | Next ]
 Nathaniel Deeth Posts: 548 Registered: 12/4/04
Re: Continuum Hypothesis Solution Posted
Posted: Apr 21, 1999 5:30 AM

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 Papus
4/15/99 Webster Kehr
4/16/99 Alan Morgan
4/18/99 Webster Kehr
4/16/99 Dave Seaman
4/16/99 Bill Taylor
4/16/99 Nathaniel Deeth
4/16/99 Jake Wildstrom
4/19/99 Sami Aario
4/16/99 Ken Cox
4/19/99 Michel Hack
4/20/99 Nathaniel Deeth
4/20/99 Ulrich Weigand
4/21/99 Nathaniel Deeth
4/21/99 Nathaniel Deeth
4/21/99 Ulrich Weigand
4/21/99 Brian David Rothbach
4/21/99 Virgil Hancher
4/22/99 Nathaniel Deeth
4/22/99 Sami Aario
4/23/99 Nathaniel Deeth
4/25/99 Sami Aario
4/22/99 Ulrich Weigand
4/23/99 Nathaniel Deeth
4/23/99 Ulrich Weigand
4/20/99 Nathaniel Deeth
4/17/99 Papus
4/19/99 Kevin Lacker
4/20/99 Bill Taylor
4/19/99 Andrew Boucher
4/19/99 Kevin Lacker
4/21/99 Andrew Boucher
4/22/99 Keith Ramsay
4/23/99 Andrew Boucher
4/24/99 Keith Ramsay
4/25/99 Andrew Boucher
4/27/99 Bill Taylor
4/27/99 David Petry
4/30/99 Keith Ramsay
5/1/99 Keith Ramsay
4/16/99 Andrew Boucher
4/16/99 Dave Seaman
4/18/99 Webster Kehr
4/19/99 Jeremy Boden
4/19/99 Sami Aario