```Date: Jan 23, 2013 4:07 PM
Author: Graham Cooper
Subject: WHAT IS WRONG WITH CANTOR'S PROOF?

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>INFINITYit's the same as arriving at the Conclusion ...  F<->not(F)From these 2 natural contradictions you Must work backwardsto find the erroneous assumption!ANTI-DIAG  =/=  ROW 1ANTI-DIAG  =/=  ROW 2ANTI-DIAG  =/=  ROW 3This 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 LIST0.200...   is MISSING FROM THE LIST0.300...   is MISSING FROM THE LIST0.110...   is MISSING FROM THE LIST0.210...   is MISSING FROM THE LIST0.310...   is MISSING FROM THE LIST0.120...   is MISSING FROM THE LIST0.220...   is MISSING FROM THE LIST0.320...   is MISSING FROM THE LIST0.102...   is MISSING FROM THE LIST0.202...   is MISSING FROM THE LIST0.302...   is MISSING FROM THE LIST0.112...   is MISSING FROM THE LIST0.212...   is MISSING FROM THE LIST0.312...   is MISSING FROM THE LIST0.122...   is MISSING FROM THE LIST0.222...   is MISSING FROM THE LIST0.322...   is MISSING FROM THE LIST0.103...   is MISSING FROM THE LIST0.203...   is MISSING FROM THE LIST0.303...   is MISSING FROM THE LIST0.113...   is MISSING FROM THE LIST0.213...   is MISSING FROM THE LIST0.313...   is MISSING FROM THE LIST0.123...   is MISSING FROM THE LIST0.223...   is MISSING FROM THE LIST0.323...   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 ofdigits in all positions!The SET OF EXHAUSTIVE ANTI-DIAGONALS   S.O.E.A.Dgets even bigger when you examine ALL PERMUTATIONSThis is not the "UNCOUNTABLY LARGE" set of infinitely long digitstrings..It's  EVERY INFINITE DIGIT STRING  of length n-1when constructed for the sublist of n rows9X9X9X9X ... n ...X9All of these strings are not MISSING from the listbecause when you construct them for 1 extra row9X9X9X9X ... n ..X9X9that SOEAD for n+1 rowscovers ALL digit strings of length n.The ANTI-DIAGONAL Methodwhen applied to the entire SET of possible stringsit can generate is 100% exhaustive!i.e.Calculate the SOEAD for a list of reals for 21 rows X 21 columnsand you get _all_ 10^20 possible digit strings of length 20.****************************************So although you CAN do Cantors proof stepCORRECTLY BY INDUCTIONBASE STEPAD_1  =/= ROW 1_1INDUCTIVE STEPAD_n  =/= ROW n_n   ->   AD_n+1 =/=  ROW_n+1_n+1BY INDUCTIONALL(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 STEPthat AD is therefore missing!But there is NO_BASE_STEPExamining 1 digit at a time....   AD is never logically missing..Herc--http://tinyurl.com/Blueprints-Hypereals
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