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Topic: CANTORS PROOF IS JUST THE INDUCTIVE STEP!
Replies: 6   Last Post: Nov 1, 2012 5:31 AM

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 Hercules ofZeus Posts: 27 Registered: 9/19/11
CANTORS PROOF IS JUST THE INDUCTIVE STEP!
Posted: Nov 1, 2012 3:36 AM

INDUCTION RULE:

P(1) & P(n)->P(S(n))
--> ALL(n) P(n)

----------------------

ANTI-DIAGONAL METHOD:

CHANGE DIGIT AD_n -> ROW n =/= AD
-> CHANGE DIGIT AD_n+1 -> ROW n+1 =/= AD

THIS IS:

P(n) -> P(S(n))

IF PROPERTY P HOLDS FOR ROW n
THEN PROPERTY P HOLDS FOR ROW n+1

TRUE!

---------------------

BUT YOU'RE MISSING THE BASE STEP!

CHANGING 1 DIGIT OF THE DIAGONAL
(eg OF COMPUTABLE REALS LIST)

-/-> (ANY ROW =/= AD)

(the permutation is arbitrary)

---------------------

TELLING US:

THE 1ST DIGIT OF THE MISSING REAL IS: 0.5

HAS 0 IMPACT!

----------------------

PROOF:

----------------------

1 IF YOU CHANGE 1 DIGIT OF THE DIAGONAL
THERE IS 0 EFFECT!

LIST OF ALL REALS
R1 0.00000..
R2 0.11211..
R3 0.22222..
R4 0.33333..
...

DIAGONAL 0.0123..
R1 0.*0000.. \
R2 0.1*211.. \
R3 0.22*22.. \
R4 0.333*3.. \
...

NEW-DIAGONAL 0.0223..
R1 0.*0000.. \
R2 0.11*11.. ^
R3 0.2*222.. v
R4 0.333*3.. \
...

NEW-PERMUTATION
R1 0.*0000.. \
R3 0.2*222.. v
R2 0.11*11.. ^
R4 0.333*3.. \
...

SAME *SET* OF REALS
R1 0.00000.. \
R3 0.22222.. \
R2 0.11211.. \
R4 0.33333.. \
...

ORIGINAL DIAGONAL = 0.0123...
CHANGED DIAGONAL = 0.0223..

-------------

CHANGING 1 DIGIT OF THE DIAGONAL

DOES NOT SATISFY ANY MISSING PROPERTY P()

ABOUT THE SET OF REALS - IT PROVES NOTHING!

-------------

Without the BASE STEP = changing 1 digit that has some predicated
restriction on the list,

there is no induction that changing all digits (one at a time) would
have any compounded effects.

Herc

Date Subject Author
11/1/12 Hercules ofZeus
11/1/12 William Elliot
11/1/12 Hercules ofZeus
11/1/12 William Elliot
11/1/12 Hercules ofZeus
11/1/12 William Elliot
11/1/12 Graham Cooper