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WHAT IS WRONG WITH CANTOR'S PROOF?
Posted:
Jan 23, 2013 4:07 PM


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!
ANTIDIAG =/= ROW 1 ANTIDIAG =/= ROW 2 ANTIDIAG =/= 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 "antidiagonals" also have this property.
Are they all missing too?
NO! they merely represent ALL STRINGS of length INFINITY1
*Given any rudimentary expressive infinite list of reals with lots of digits in all positions!
The SET OF EXHAUSTIVE ANTIDIAGONALS 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 n1 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 ANTIDIAGONAL 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/BlueprintsHypereals



