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Topic: RE: Matheology 400: Quantifier Confusion
Replies: 188   Last Post: Dec 13, 2013 5:48 AM

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fom

Posts: 1,968
Registered: 12/4/12
Re: Matheology 400: Quantifier Confusion
Posted: Dec 6, 2013 9:21 PM
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On 12/6/2013 3:11 PM, WM wrote:
> Am Freitag, 6. Dezember 2013 20:11:32 UTC+1 schrieb Michael F. Stemper:
>

>>
>> (For starters, did
>>
>> Cantor include the assumption that the list had all reals?)

>
> No. He said: Ist E1, E2, ..., E?, ... irgendeine einfach unendliche Reihe von Elementen der Mannigfaltigkeit M
> Let ... be any infinite sequence of elements ...
>


You are correct with regard to the 1874 paper.
Ewald has his earlier statements to Dedekind
of 1873 phrased differently.

The paper has the proof previously posted by
Virgil:

<begin quote>


> *********************************************
> A PROOF OF THE UNCOUNTABILITY OF THE REALS
> (A variation on Cantor's FIRST proof)
>
> ASSUMPTIONS:
>
> (1) the intersection of a strictly nested sequence of closed real
> intervals (the endpoints of each interval being interior points of the
> previous interval) is not empty.
>
> (2) A strictly increasing sequence of naturals does not have a natural
> as its limit,
>
> (3a) A strictly increasing but bounded sequence of reals has a real
> number as a limit, its least upper bound, different from every member of
> the sequence.
>
> (3b) A strictly decreasing but bounded sequence of reals has a real
> number as a limit, its greatest lower bound, different from every member
> of the sequence.
>
> Proof:
>
> If the set of all reals is countable then we may assume each real can be
> and has been paired with a natural so that different reals are paired
> with different naturals with none of either left out.
>
> Assuming this has been done, take the two reals corresponding to the
> lowest naturals as endpoints of a real interval.
>
> It is clear that all the interior points of this real interval must be
> paired with naturals larger than those naturals paired with its
> endpoints.
>
> Now take the two reals INTERIOR to the previous interval with the lowest
> naturals to be the endpoints of a subinterval of that interval.
>
> It is clear that the interior points of such a real interval must be
> paired with naturals larger than the naturals paired with its endpoints.
>
> By repeating this process of forming new intervals one generates a
> decreasing, but never empty, sequence of closed real intervals each of
> which contains only points with higher attached naturals than its
> endpoints have.
>
> The intersection of such a nested sequence of closed intervals is not
> empty, but the natural associated with any of its members is necessarily
> larger than all of the infinitely many natural numbers associated with
> those infinitely many endpoints.
>
> But there is no natural number larger than infinitely many different
> natural numbers.
>
> This is a contradiction which can only have been caused by our original
> assumption that the reals were countable, so it proves they are not
> countable.
>
> QED!
>
> NOTE: While I have never seen this particular proof in the literature of
> countability, it is so obvious that I doubt that it is original with me.
>


<end quote>



Date Subject Author
12/4/13
Read RE: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/4/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/4/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/4/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/4/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/4/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/6/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/6/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/7/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/8/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/8/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/6/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/7/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/7/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/7/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/8/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/8/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/6/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/7/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/4/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/4/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/5/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/5/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Michael F. Stemper
12/5/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/6/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/6/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Michael F. Stemper
12/6/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/7/13
Read Re: Matheology 400: Quantifier Confusion
albrecht
12/7/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/7/13
Read Re: Matheology 400: Quantifier Confusion
albrecht
12/6/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/7/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/7/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/7/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/7/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/8/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/8/13
Read Re: Matheology 400: Quantifier Confusion
Ralf Bader
12/8/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/6/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/6/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/7/13
Read Re: Matheology 400: Quantifier Confusion
ross.finlayson@gmail.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/6/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/6/13
Read Re: Matheology 400: Quantifier Confusion
albrecht
12/6/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/6/13
Read Re: Matheology 400: Quantifier Confusion
albrecht
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/6/13
Read Re: Matheology 400: Quantifier Confusion
albrecht
12/6/13
Read Re: Matheology 400: Quantifier Confusion
albrecht
12/6/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/7/13
Read Re: Matheology 400: Quantifier Confusion
albrecht
12/8/13
Read Re: Matheology 400: Quantifier Confusion
albrecht
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/5/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/5/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/4/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/5/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/5/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/6/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/6/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/5/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/5/13
Read Re: Matheology 400: Quantifier Confusion
ross.finlayson@gmail.com
12/9/13
Read Re: Matheology 400: Quantifier Confusion
William Hughes
12/9/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/9/13
Read Re: Matheology 400: Quantifier Confusion
William Hughes
12/9/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/9/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/10/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/10/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/10/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/10/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/10/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Ralf Bader
12/10/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/9/13
Read Re: Matheology 400: Quantifier Confusion
William Hughes
12/10/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/10/13
Read Re: Matheology 400: Quantifier Confusion
William Hughes
12/10/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/10/13
Read Re: Matheology 400: Quantifier Confusion
William Hughes
12/10/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/10/13
Read Re: Matheology 400: Quantifier Confusion
William Hughes
12/10/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/11/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/11/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/11/13
Read Re: Matheology 400: Quantifier Confusion
William Hughes
12/11/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/11/13
Read Re: Matheology 400: Quantifier Confusion
William Hughes
12/11/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/11/13
Read Re: Matheology 400: Quantifier Confusion
William Hughes
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/11/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/13/13
Read Re: Matheology 400: Quantifier Confusion
albrecht
12/10/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/10/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/10/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/10/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/11/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/11/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/11/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/12/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/12/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/12/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/12/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/12/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/12/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/12/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/11/13
Read Re: Matheology 400: Quantifier Confusion
fom
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/12/13
Read Re: Matheology 400: Quantifier Confusion
wolfgang.mueckenheim@hs-augsburg.de
12/12/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/11/13
Read Re: Matheology 400: Quantifier Dyslexia
Virgil
12/11/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/10/13
Read Re: Matheology 400: WM's Quantifier Confusion
Virgil
12/11/13
Read Re: Matheology 400: WM's Quantifier Confusion
William Hughes
12/11/13
Read Re: Matheology 400: WM's Quantifier Confusion
Virgil
12/11/13
Read Re: Matheology 400: WM's Quantifier Confusion
William Hughes
12/11/13
Read Re: Matheology 400: WM's Quantifier Confusion
Virgil
12/11/13
Read Re: Matheology 400: WM's Quantifier Confusion
William Hughes
12/11/13
Read Re: Matheology 400: WM's Quantifier Confusion
fom
12/11/13
Read Re: Matheology 400: WM's Quantifier Confusion
Virgil
12/10/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/9/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/9/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/9/13
Read Re: Matheology 400: Quantifier Confusion
William Hughes
12/9/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/9/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/9/13
Read Re: Matheology 400: Quantifier Confusion
William Hughes
12/9/13
Read Re: Matheology 400: Quantifier Confusion
Virgil
12/9/13
Read Re: Matheology 400: Quantifier Confusion
Tucsondrew@me.com
12/10/13
Read Re: Matheology 400: Quantifier Confusion
William Hughes

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