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Topic: THE FOUNDATION OF NUMBERS BY CANTOR!
Replies: 6   Last Post: Jun 2, 2012 4:33 PM

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netzweltler

Posts: 304
From: Germany
Registered: 8/6/10
Re: THE FOUNDATION OF NUMBERS BY CANTOR!
Posted: Jun 1, 2012 2:07 AM
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On 31 Mai, 05:38, Graham Cooper <grahamcoop...@gmail.com> wrote:
> THE FOUNDATION OF NUMBERS BY CANTOR!
>
> AD[r]=/=LIST[r,r] -> AD[r]=/=LIST[r,r]
>
> -> 2^aleph_0 > aleph_0
>
> -> 2x2x2x2... > 1+1+1+1...
>
> Incomplete..Inconsistent..Uncomputable..Uncountable..Unformalizable..
> Unspecifiable..NotUniversal..NotVerifiable
>
> Gee  I  W o n d e r  why that is!!??
>
> Herc


As far as I know 2 x 2 x 2 x ... actually means the infinite cartesian
product {0,1} x {0,1} x {0,1} x ...
If you calculate this product step by step (in countably infinitely
many steps) you will NOT get the set of all infinite binary sequences.
As you can see here (intermediate results of the calculations shifted
to the right):

Step 1:

...00000 0
...00000 1
...000010
...000011
...000100
...000101
...000110
...000111
...001000
...

Step 2:

...0000 00
...0000 01
...0000 10
...0000 11
...000100
...000101
...000110
...000111
...001000
...

Step 3:

...000 000
...000 001
...000 010
...000 011
...000 100
...000 101
...000 110
...000 111
...001000
...

and so on, you simply get the infinite binary sequences of the natural
numbers (if ...000001 = 001 = 01 = 1).

--
netzweltler




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