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Topic: WHAT IS WRONG WITH CANTOR'S PROOF?
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Graham Cooper

Posts: 4,319
Registered: 5/20/10
WHAT IS WRONG WITH CANTOR'S PROOF?
Posted: Jan 23, 2013 4:07 PM
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DIGIT 1 IS DIFFERENT TO LIST[1,1]
DIGIT 2 IS DIFFERENT TO LIST[2,2]
DIGIT 3 IS DIFFERENT TO LIST[3,3]
...
AND SO ON..


There is no "and so on..." in formal mathematics!

When you arrive at the Conclusion ... X>INFINITY
it's the same as arriving at the Conclusion ... F<->not(F)

From these 2 natural contradictions you Must work backwards
to find the erroneous assumption!

ANTI-DIAG =/= ROW 1
ANTI-DIAG =/= ROW 2
ANTI-DIAG =/= ROW 3

This will only hold IFF it holds by INDUCTION over N



*************************

A SUBLIST OF REALS IN [BASE 4]
R1 0.0000...
R2 0.3333...
R3 0.3210...
...


0.100... is MISSING FROM THE LIST
0.200... is MISSING FROM THE LIST
0.300... is MISSING FROM THE LIST
0.110... is MISSING FROM THE LIST
0.210... is MISSING FROM THE LIST
0.310... is MISSING FROM THE LIST
0.120... is MISSING FROM THE LIST
0.220... is MISSING FROM THE LIST
0.320... is MISSING FROM THE LIST
0.102... is MISSING FROM THE LIST
0.202... is MISSING FROM THE LIST
0.302... is MISSING FROM THE LIST
0.112... is MISSING FROM THE LIST
0.212... is MISSING FROM THE LIST
0.312... is MISSING FROM THE LIST
0.122... is MISSING FROM THE LIST
0.222... is MISSING FROM THE LIST
0.322... is MISSING FROM THE LIST
0.103... is MISSING FROM THE LIST
0.203... is MISSING FROM THE LIST
0.303... is MISSING FROM THE LIST
0.113... is MISSING FROM THE LIST
0.213... is MISSING FROM THE LIST
0.313... is MISSING FROM THE LIST
0.123... is MISSING FROM THE LIST
0.223... is MISSING FROM THE LIST
0.323... is MISSING FROM THE LIST

**********************

HINT: DIGIT 1 IS DIFFERENT TO LIST[1,1]
HINT: DIGIT 2 IS DIFFERENT TO LIST[2,2]
HINT: DIGIT 3 IS DIFFERENT TO LIST[3,3]


====================

But 9X9X9X9X.....
other "anti-diagonals" also have this property.

Are they all missing too?

NO! they merely represent ALL STRINGS of length INFINITY-1

*Given any rudimentary expressive infinite list of reals with lots of
digits in all positions!

The SET OF EXHAUSTIVE ANTI-DIAGONALS S.O.E.A.D
gets even bigger when you examine ALL PERMUTATIONS

This is not the "UNCOUNTABLY LARGE" set of infinitely long digit
strings..

It's EVERY INFINITE DIGIT STRING of length n-1
when constructed for the sublist of n rows

9X9X9X9X ... n ...X9

All of these strings are not MISSING from the list
because when you construct them for 1 extra row

9X9X9X9X ... n ..X9X9

that SOEAD for n+1 rows
covers ALL digit strings of length n.

The ANTI-DIAGONAL Method
when applied to the entire SET of possible strings
it can generate is 100% exhaustive!

i.e.
Calculate the SOEAD for a list of reals for 21 rows X 21 columns
and you get _all_ 10^20 possible digit strings of length 20.


****************************************

So although you CAN do Cantors proof step
CORRECTLY BY INDUCTION

BASE STEP
AD_1 =/= ROW 1_1

INDUCTIVE STEP
AD_n =/= ROW n_n -> AD_n+1 =/= ROW_n+1_n+1

BY INDUCTION
ALL(n) AD_n =/= ROW_n_n


*******************************

The END RESULT is not obtainable By_Induction!

ALL(n) AD =/= ROW_n

===================

"AND SO ON..." is shorthand for the INDUCTIVE STEP

that AD is therefore missing!

But there is NO_BASE_STEP

Examining 1 digit at a time.... AD is never logically missing..



Herc
--
http://tinyurl.com/Blueprints-Hypereals





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