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

 Messages: [ Previous | Next ]
 Ken Cox Posts: 339 Registered: 12/12/04
Re: Continuum Hypothesis Solution Posted
Posted: Apr 16, 1999 6:07 PM

Nathan the Great wrote:
> Contrary to the dogma of The Dark Master, a function f: N -> R exists. The
> following construction algorithm produces EVERY decimal digit string in the Real
> interval [0,1). Note, because this is a NEVER ENDING algorithm that keeps
> churning out numbers to greater and greater digit lengths, unbounded (aka
> infinite) length decimal strings are in essence constructed.
>
> Number\$(1) = "0." ;first number in the interval [0,1).
> Reference = 1 ;points to a pre-extended number string.
> NewIndex = 2 ;point where new (extended) numbers get appended.
>
> Do
> For Digit = 0 to 9
> Number\$(NewIndex) = Number\$(Reference) & Digit
> NewIndex = NewIndex + 1
> Next Digit
> Reference = Reference + 1
> Loop

This algorithm does not produce any of the irrational numbers
in [0,1), such as 1/sqrt(2) or pi-3. Thus it is not a function
from N to R.

Proof: Assume to the contrary that the algorithm produces some
irrational number. Then this number must be at some particular
slot K in the table. However, the number in slot K has no more
than K digits (actually it's O(log10 K), but we don't need that
tight a bound). But the decimal expansion of our irrational
number is longer than K, or the number would be rational. Thus
the number does not appear in the table. As this contradicts
our assumption that it is in the table, we must conclude that
the assumption is wrong. The irrational is thus not in the table.

--
Ken Cox kcc@research.bell-labs.com

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