
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 neverending 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 ErlangenNuernberg, > Martensstr. 3, D91058 Erlangen, Phone: +49 9131 857688
Mr. Ulrich, I have two questions:
(1) Is Cantor's Diagonal Number a completed infinity? (2) Does the "onefellswoop" 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 onefellswoop 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 preexisting 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

