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The MYTH of TRANSFINITE SIZED SETS!
Posted:
Jan 12, 2013 4:13 AM
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Consider this Dialog with a Transfinite Theorist CAM who believes
Cantor's proof is that there are more than the amount of natural
numbers of real numbers...
TOM: I can describe part of my infinite list of real numbers..
CAM: OK go on!
TOM: The first real begins 0.5...
CAM: I can describe 9 missing reals!
TOM: OK go on!
CAM:
A real beginning 0.0... is missing from your list!
A real beginning 0.1... is missing from your list!
A real beginning 0.2... is missing from your list!
A real beginning 0.3... is missing from your list!
A real beginning 0.4... is missing from your list!
A real beginning 0.6... is missing from your list!
A real beginning 0.7... is missing from your list!
A real beginning 0.8... is missing from your list!
A real beginning 0.9... is missing from your list!
TOM: Don't be absurd! How could you know what was missing from my
infinite list?
CAM: I changed the first digit of the first real in your list!
A real beginning 0.0... is different than your 1st real!
A real beginning 0.1... is different than your 1st real!
A real beginning 0.2... is different than your 1st real!
A real beginning 0.3... is different than your 1st real!
A real beginning 0.4... is different than your 1st real!
A real beginning 0.6... is different than your 1st real!
A real beginning 0.7... is different than your 1st real!
A real beginning 0.8... is different than your 1st real!
A real beginning 0.9... is different than your 1st real!
TOM: OK here is 1 more digit from my list. X is currently unknown.
0.5...
0.X3...
CAM: I can describe roughly 81 missing reals from your list!
TOM: OK here are 2 more digits from my list to add!
0.56...
0.73...
CAM: Now I can describe nearly 162 missing reals from this list
0.56..
0.73..
and this list
0.73...
0.56...
*****************************
In fact, CAM can show up to ( barring duplicates )
9*9*9*9*9....*9 *N*N-1*N-2*....*2*1
MISSING REALS just by perusing the first N digits of the first N
reals!
By the Pigeon Hole principle, this is MAGNITUDES GREATER
than the total possible number of digit combinations possible with N
digits!
In Fact, give a medium sized list of reals you could fit in a sheet of
paper
ALL DIGIT SEQUENCES ARE *MISSING* !
Herc
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