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Topic: Why does Cantor's diagonal argument fail?
Replies: 5   Last Post: Jan 18, 2014 4:14 PM

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ross.finlayson@gmail.com

Posts: 908
Registered: 2/15/09
Re: Why does Cantor's diagonal argument fail?
Posted: Jan 18, 2014 4:14 PM
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On 1/18/2014 9:52 AM, Ross A. Finlayson wrote:
> On 1/18/2014 8:31 AM, mueckenh@rz.fh-augsburg.de wrote:
>> On Saturday, 18 January 2014 14:53:43 UTC+1, Ben Bacarisse wrote:
>>> mueckenh@rz.fh-augsburg.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
>> 9-problem. He did not use numbers but nevertheless should have
>> mentioned it. The 9-problem 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 rationals-complete 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 well-known as a HUGE pan-European
(in those days the center) crank.

This is somewhat vindicated as his notation is regular
today for a central feature of mathematics: real analysis.




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