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Topic: Uncountability of the Real Numbers Without Decimals
Replies: 110   Last Post: Dec 10, 2013 4:33 AM

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 Tucsondrew@me.com Posts: 1,161 Registered: 5/24/13
Re: Uncountability of the Real Numbers Without Decimals
Posted: Dec 5, 2013 12:07 PM

On Thursday, December 5, 2013 3:35:14 AM UTC-7, WM wrote:
> Am Montag, 2. Dezember 2013 21:41:33 UTC+1 schrieb Zeit Geist:
>

> > > The set of positive rational numbers that is less than the natural number n and has not been enumerated by the first n natural numbers grows with n. It is impossible eneumerate all rational numbers, i.e., to remove all rationals from the state of being not enumerated to the state of being enumerated.
>
> > Impossible? How about a proof.
>
> Proof (1): In order enumerate a rational, you have to take (identify) it and map it on a natural number. Cantor tries so, but fails, since each of his naturals belongs to a finite initial segment of |N whereas infinitely many cannot be taken (identified).
>

A function, L: N --> Q, which is Surjective ( This means A( q e Q )( E( n e N )( L(n) = q ) ). ) is Sufficient to show that q can be enumerated.

Your "You can't Count to Infinity" does not Invalidate the fact that the Function is Not enough.
The Function does the Counting.

> Contradict this proof by showing one natural number that is not followed by infinitely many or has at least as many predecessors.

It doesn't matter how may follow, as long as they all are accounted for at Some point.

> Proof (2): The putative enumeration is a super task:
>
> M(1) = { }
>
> M_(n+1) = (M_n U {rationals in (n-1, n]}) \ q_n
>
> The set of rationals that have not been enumerated by one of the first n naturals but are less than n is infinite for every n in |N and does not decrease for every n. The belief in an enumeration of all rationals implies that lim M_n = { }. Contradiction.
>

Same Failing Refutation.

> > > Here the intentional confusion of "for every n in |N" and "for all n in |N" has been used. It has been neglected that beyond every n there are infinitely many following, such that never all can have been used.
>
> > And actually, when writing it I knew you would say that.
>
> Did you understand it? Or what is your counter argument?
>

Hmmm. Let's see. You say We use "for every n in N" implies "for all n in N". Well, We do, because it does. Do you need the Proof again?

Never, you won't comprehend it anyway.

> > You are welcome to work your own Mathematical System sans the AoI.
>
> On the contrary, AoI says just what I say, namely that every n is followed by infinitely many.
>

And that is Irrelevant.

> The mistake lies only in the false interpretation of "set" as an actually existing object. Not: There is no definition of "set". Therefore your interpretation has not basis.
>

I have said before these are Idealized Mathematical Objects. You claim the are Actual by insisting we can't "count" an Infinite Set.

> Regards, WM

ZG

Date Subject Author
12/2/13 Tucsondrew@me.com
12/2/13 William Elliot
12/2/13 Tucsondrew@me.com
12/2/13 G. A. Edgar
12/2/13 Tucsondrew@me.com
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 gnasher729
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 Virgil
12/2/13 Tucsondrew@me.com
12/2/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Virgil
12/3/13 Virgil
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/5/13 Virgil
12/10/13 Robin Chapman
12/2/13 Virgil
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Virgil
12/2/13 Tucsondrew@me.com
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/5/13 Michael F. Stemper
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Tucsondrew@me.com
12/6/13 wolfgang.mueckenheim@hs-augsburg.de
12/6/13 Virgil
12/6/13 Brian Q. Hutchings
12/7/13 Brian Q. Hutchings
12/7/13 Brian Q. Hutchings
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 fom
12/7/13 albrecht
12/7/13 fom
12/7/13 ross.finlayson@gmail.com
12/8/13 albrecht
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 fom
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 fom
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/7/13 fom
12/8/13 Virgil
12/7/13 Virgil
12/7/13 Virgil
12/7/13 Virgil
12/8/13 albrecht
12/6/13 Virgil
12/6/13 Virgil
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/5/13 Virgil
12/3/13 Virgil
12/3/13 Michael F. Stemper
12/3/13 Virgil
12/3/13 fom
12/2/13 Tucsondrew@me.com
12/2/13 wolfgang.mueckenheim@hs-augsburg.de
12/2/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Virgil
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Virgil
12/2/13 Virgil
12/2/13 Tucsondrew@me.com
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 gnasher729
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 gnasher729
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Virgil
12/3/13 Virgil
12/3/13 Virgil
12/5/13 gnasher729
12/3/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/5/13 Tucsondrew@me.com
12/5/13 Virgil
12/3/13 wolfgang.mueckenheim@hs-augsburg.de
12/3/13 Tucsondrew@me.com
12/5/13 wolfgang.mueckenheim@hs-augsburg.de
12/5/13 Tucsondrew@me.com
12/7/13 wolfgang.mueckenheim@hs-augsburg.de
12/7/13 Virgil
12/8/13 wolfgang.mueckenheim@hs-augsburg.de
12/8/13 Virgil
12/5/13 Virgil
12/3/13 Virgil
12/2/13 ross.finlayson@gmail.com
12/4/13 ross.finlayson@gmail.com
12/3/13 albrecht
12/3/13 Tucsondrew@me.com
12/5/13 albrecht
12/5/13 Tucsondrew@me.com