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Re: Continuum Hypothesis Solution Posted
Posted:
Apr 21, 1999 9:29 AM
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Nathan the Great <mad@ashland.baysat.net> writes:
> (1) Is Cantor's Diagonal Number a completed infinity?
It is one specific real number. Depending on how you define real numbers,
you'll need an infinite amount of information to specify a real number,
however. E.g. if you define real numbers by Dedekind cut, every real
number is itself an infinite set.
I'm not exactly sure what you mean by 'completed' infinity, though.
> (2) Does the "one-fell-swoop" in the proof below terminate?
Again, please specify exactly what you mean by 'one-fell-swoop'.
If you somehow interpret this as an algorithm of some kind, of course
it does not terminate.
Nevertheless, if you interpret is an a method of specifying a certain
real number, it is valid as such.
[snip]
> 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.
This specification defines a certain real number (namely, pi). If you
choose to interpret a real number as 'completed infinity' then this is so.
>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).
This makes no sense. You again confuse different things. On the one hand,
you have a sequence of rational numbers defined by your construction.
Each of these is, well, rational. But none of these is equal to NDN.
NDN is the real number specified as limit of the sequence of these
rational numbers. As such, it may be rational or irrational; you'll
have to examine the number itself to find out which is the case (and
of course, in this particular case, the number is irrational).
--
Ulrich Weigand,
IMMD 1, Universitaet Erlangen-Nuernberg,
Martensstr. 3, D-91058 Erlangen, Phone: +49 9131 85-7688
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