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::::: "...but that new anti-diagonal number was included in the list when ''c'' was the anti-diagonal number." No it wasn't! [[User:Ossi|Ossi]] 19:47, 8 September 2007 (UTC)
:::::: It is wise to understand that any real number is a constant [that is, in place-value positional numeral system like decimal or binary, its every digit is known (or could be computed “given sufficient computing time and space resources” which is a standard axiom in the modern theory of computation)] so they have fixed existence (unlike a variable) and there could not be any algorithm such as Cantor’s anti-diagonalization argument that can produce “new” numbers (that is, not already on the row-list unless it is an extended number such as row-listing all the rationals and having irrational diagonal and anti-diagonal number or row-listing all the real numbers and having complex diagonal and anti-diagonal number) like the 2nd anti-diagonal number, say D, when the first anti-diagonal number C is in the row-list. Any real number is either the anti-diagonal number or in the “actually incomplete” row-listing. If D was not included in the row-listing when C was the anti-diagonal number, then you don’t have to call upon the excluded anti-diagonal number C because your first row-list is already incomplete since it also excludes another real number D. In the sequence of ALL real numbers {X<sub>n</sub>} = {Y<sub>n</sub>} U {A<sub>n</sub>} U {B<sub>n</sub>} U {C} of my discussion above, C is an arbitrary real number limit. If the 2nd limit D is not equal to the first limit C, then D is in {Y<sub>n</sub>} or in {A<sub>n</sub>} or in {B<sub>n</sub>} when C is the limit, and vice versa on “C” and “D” — otherwise, {X<sub>n</sub>} does not enumerate all the real numbers as presupposed.
:::::: From a different perspective, the following are tautologous diagonalization (not anti-diagonalization like Cantor’s) arguments:
::::::: * a truly complete row-listing with 1st row the positive integers in their respective columns; in row 2, the respective positive integers 2nd powers (squares) ; in row 3, the respective positive integers 3rd powers (cubes); …; in row n, the respective positive integers nth powers; … . Then, clearly, the diagonal sequence of n<sup>n</sup> is not in the row-list (Leo Zippin in “Uses of Infinity” [MAA, 1962] claims this to be an example of Cantor’s diagonal argument);
::::::: * a truly complete enumeration or row-listing of all the fractional rational numbers would have irrational diagonal and anti-diagonal numbers;
::::::: * a truly complete enumeration or row-listing of all the fractional real numbers — which consists of all the rational (nonterminating but periodic binary or decimal expansion) and irrational (nonterminating and nonperiodic expansion) fractional numbers with each collection exhausting the positive integer row-indices and the representation 0.b<sub>1</sub>b<sub>2</sub>b<sub>3</sub>… (for the rational numbers, there is no limit in the period-length while for the irrational numbers, each has a nonzero digit at the omegath position due to the 1-1 correspondence, say π-3 in binary, <0.0, 0.00, 0.001, 0.0010, 0.00100, 0.001001, …, π-3> <--> <0, 0, 1, 0, 0, 1, …, 1<sub>ω</sub>) — would simply have irrational anti-diagonal (with 1<sub>ω</sub> if diagonal is rational (without or with 0<sub>ω</sub>), or vice versa on “ratiobal” and “irrational”;
::::::: * a truly complete enumeration or row-listing of all the fractional algebraic (inclined measurements) real numbers would have transcendental (curved, compounded continuously measurements) real diagonal and anti-diagonal numbers — and vice versa on “algebraic” and “transcendental”;
::::::: * a truly complete row-listing of all the fractional real numbers (complex numbers with no imaginary part — both algebraic and transcendental) would have complex (with real and imaginary part) diagonal and anti-diagonal numbers;
::::::: * a truly complete enumeration or row-listing of all the fractional complex numbers would have quaternion diagonal and anti-diagonal numbers;
::::::: * etc.
:::::: Even Ludwig Wittgenstein missed my point here -- you might want to read [http://www.math.ucla.edu/%7easl/bsl/0401-toc.htm “An Editor Recalls Some Hopeless Papers”] by Wilfrid Hodges. The very simple flaw in Hodges’ presented variant of Cantor’s anti-diagonal argument which claims to prove the “uncountability” of all the real numbers by “demonstrating” that there could not be any 1-to-1 correspondence between all the natural numbers and all the fractional real numbers is the common false belief that the arbitrary decimal place-value positional numeral system representation 0.d<sub>1</sub>d<sub>2</sub>d<sub>3</sub>…d<sub>n</sub>…, where d<sub>n</sub> is in {0,1,2,3,4,5,6,7,8,9} for every natural number n, denotes the fractional expansion of either a rational (nonterminating, periodic) or an irrational (nonterminating, nonperiodic) real number between 0 and 1 when indeed just the rational numbers (since they have no period-length limit — in other words, any finite sequence of digits is a possible period) already exhaust them (on the other hand, the irrational numbers require the representation 0.d<sub>1</sub>d<sub>2</sub>d<sub>3</sub>…d<sub>n</sub>…1<sub>ω</sub> due to the 1-1 correspondence, say π – 3 = 0. 0010010000111... in binary system, <0.0, 0.00, 0.001, 0.0010, 0.00100, 0.001001, …, π - 3> <--> <0, 0, 1, 0, 0, 1, …, 1<sub>ω</sub>. [BenCawaling@Yahoo.com] [[User:BenCawaling|BenCawaling]] ([[User talk:BenCawaling|talk]]) 06:25, 26 March 2008 (UTC)
:"Cantor's first proof" is new to me, and I have to say it's delightful. I agree that mathematicians generally believe the diagonal argument to be Cantor's first. However, I'm not completely convinced that this isn't really a diagonal argument in disguise. I need to think about this a bit. [[User:Dmharvey|Dmharvey]] [[Image:User_dmharvey_sig.png]] [[User talk:Dmharvey|Talk]] 22:45, 6 Jun 2005 (UTC)
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