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Math Forum » Discussions » sci.math.* » sci.math

Topic: CANTORS PROOF DEMO the Reals are UN-COUNTABLE
Replies: 2   Last Post: Jan 22, 2013 9:00 PM

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Graham Cooper

Posts: 4,348
Registered: 5/20/10
CANTORS PROOF DEMO the Reals are UN-COUNTABLE
Posted: Jan 9, 2013 7:34 PM
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A SUBLIST OF REALS IN [BASE 4]

R1 0.0000...
R2 0.3333...
R3 0.3210...
...

DIAGONAL = 0.031...

DEFINE
AD(d) = 2 IFF DIAGONAL(d) < 2
AD(d) = 1 IFF DIAGONAL(d) > 1

AD=0.212... is MISSING FROM THE LIST

PROOF
DIGIT 1 (2) IS DIFFERENT TO LIST[1,1] (0)
DIGIT 2 (1) IS DIFFERENT TO LIST[2,2] (3)
DIGIT 3 (2) IS DIFFERENT TO LIST[3,3] (1)
AND SO ON

So AD is DIFFERENT to EVERY ROW
since This Holds For Any Arbitrary List Of Reals
there is a missing Real for any List Of Reals
therefore Reals are Un-Countable!

Herc

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