Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

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

 Search Thread: Advanced Search

 Messages: [ Previous | Next ]
 Virgil Posts: 8,833 Registered: 1/6/11
Re: Uncountability of the Real Numbers
Posted: Dec 3, 2013 5:24 PM
 Plain Text Reply

In article <4222f57d-3cc9-4c94-94db-cc83f21f0536@googlegroups.com>,
WM <wolfgang.mueckenheim@hs-augsburg.de> wrote:

> Am Dienstag, 3. Dezember 2013 11:21:46 UTC+1 schrieb christian.bau:
>

> > Shouldn't really replying at all
>
> Yes, if you prefer to stay blind, you should only talk to other blinds.

Being blind to what is not really there is not a drawback,
but WM's seeing things which are not there is.
>
> >"Zeit Geist"s fine post proved that the real numbers are uncountable by
> >starting with an assumption that there is a sequence containing them all
> >(which is _by definition_ the opposite of the reals being uncountable) and
> >went on to demonstrate a contradiction, which means the assumption of the
> >existence of such a sequence must be false.

>
> It seems you have not yet understood the main point. To contradict a
> nonsensical definition does not prove anything: There is no sequence
> containing "all" rational numbers.

Yes, there are such sequences, I have even constructed some of them
myself.

Consider this well-ordering of |Q,
where each rational in |Q is represented by m/n, with m an integer,
n a natural. and with m and n having no common factor greater than 1,
and ordered as follows:
if |m'| + n' < |m"| + n" then m'/n' precedes m"/n"
if |m'| + n' = |m"| + n" and m' < m" then m'/n' precedes m"/n"

So 0/1 comes first,
then -1/1, and 1/1,
then -2/1, -1/2, 1/2 and 2/1,
and so on.

If WM claims that this ordering is NOT a well-ordering of all the
rationals with a unique non-successor element or that it is NOT
order-isomorphic to the naturally ordered naturals, let him try to prove
it.

He will fail.

A little thought, if WM is capable of it, should show him that every
rational thus ordered has an immediate successor and everyone other than
0/1 has an immediate predecessor.

>
> This *would* be the point if such sequence could exist at all. But that is
> impossible.

In what respect is the sequencing of rationals described above
"impossible"?

Please try at least to understand that you are mistaken if you
> think that you can conclude from
>

>
> The full text of the first line reads
>
> "every rational number can be enumerated if and only if beyond that
> rational number infinitely many are following."
>
> Can you understand that it is impossible to enumerate any rational number
> violating this requirement?

I do not see that there is any rational number which that rule prevents
from being correctly ordered. When one has a finite rule, like the one I
posted above, it can be applied in infinitely many instances without
getting worn out or used up.

> But if countability *would* be a sensible notion, then there is a simple
> proof that all definable reals belong to a countable set. Since all
> "uncountability proofs" prove the existence of a further definable number,
> all are trash.

Only if one assumes up front that that is the case.

Absent WM's a priori assumption, there is no proof that any such
assumption is warranted.
--

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

© The Math Forum at NCTM 1994-2017. All Rights Reserved.