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Re: Why does Cantor's diagonal argument fail?
Posted:
Jan 18, 2014 4:14 PM


On 1/18/2014 9:52 AM, Ross A. Finlayson wrote: > On 1/18/2014 8:31 AM, mueckenh@rz.fhaugsburg.de wrote: >> On Saturday, 18 January 2014 14:53:43 UTC+1, Ben Bacarisse wrote: >>> mueckenh@rz.fhaugsburg.de writes: >>> >>> <snip> >>> >>>> Of course. I have been quoting an exceptionally simple example for >>>> years: >>> >>>> >>> >>>> 0.0 >>> >>>> 0.1 >>> >>>> 0.11 >>> >>>> 0.111 >>> >>>> ... >>> >>>> >>> >>>> If the antidiagonal is constructed by the replacement of 0 by 1 then >>> >>>> list and diagonal and antidiagonal are all well defined: The list as >>> >>>> the sequence of finite approximations of 1/9, the diagonal as zero, >>> >>>> and the antidiagonal as 1/9. >>> >>> >>> >>> ...which differs from all entries. >> >> Of course, but not at a finite place, i.e., not by an indexed digit. >> They are all in the list by definition and by possibility proven by >> Cantor. The limit is one of several numbers being defined by a finite >> rule. The set of this numbers is not existing as a completed set, >> because we always can make new definitions. But we know that we never >> can make uncountable many definitions. For that sake we would need at >> least one word with uncountably many different finite definitions. But >> how so? Even all finite subsets of the set of natural numbers belong >> to a countable set. In order to get an uncountable power set, we need >> infinite subsets. Alas, they cannot be defined by their infinitely >> many elements but only by finite rules. >> >> By the way Cantor did not think of defined lists and limits. He argued >> that a number that differs from every list number by at least one >> digit cannot be in the list. He did not think of a complete list >> (although he had proven its possibility). He did not even think of the >> 9problem. He did not use numbers but nevertheless should have >> mentioned it. The 9problem was the first hint that showed that limits >> can be in the list although all digits differ. It has been removed by >> the first one who recognized it. (I don't know who was the first. >> Fraenkel, Jourdain?) The problem of the rationalscomplete list had >> not been considered, as far as I know, before I did. But perhaps >> someone can show that it has been mentioned before. I discovered its >> simplest form in Nov. 2004. >> >> In case noone has ever mentioned it, would that mean that they did not >> do so because it was obviously cranky? >> >> Regards, WM >> > > > > > Leibniz at least has constructions the same. Anything so > simple you well could imagine could be known to thinkers. > > Leibniz specifically has a name for this function. > Obviously I frame this context as of "the equivalency > function", here for that the naturals, and the reals in > R[0,1], as the range of this function, are equivalent. > > Also many modern theories have these objects with their > names in the theory. EF is a function in terms of N and R, > this makes it simple to interpret. > >
Of course, Leibniz is wellknown as a HUGE panEuropean (in those days the center) crank.
This is somewhat vindicated as his notation is regular today for a central feature of mathematics: real analysis.



