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:::::: 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
:From the constructionists point view Cantors proof '''is''' "flawed". Let me first say that that I am '''not''' a constructionist, and that I do accept Cantor proof in full. Constructionists don't even consider it proven that there is no greatest number, and counterarguments won't bite. Here is why: Suppose there '''is''' a greatest natural number and call it Max. Then a kid in kindergarten can show that there is a number SuperMax = Max + 1 that is greater than Max. Contradiction right? '''Wrong''', says the constructionist, because you haven't shown Max to exist, and therefore you '''cannot''' use it in any way in a proof of anything. Reductio ad absurdum arguments aren't generally accepted. Constructionists are "more rigorous" than most other mathematicians, probably in a sense that could be quantified exactly if the subset of the aximoms of ZFC that they accept is specified. The Axiom of Infinity is obviously not one of them. By the same token they can "proove fewer theorems" than most other mathematicians. The uncountability of the reals is obviously one of them. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 02:20, 30 November 2007 (UTC)
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