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

 Messages: [ Previous | Next ]
 Virgil Posts: 8,833 Registered: 1/6/11
Re: Uncountability of the Real Numbers Without Decimals
Posted: Dec 5, 2013 7:38 PM

WM <wolfgang.mueckenheim@hs-augsburg.de> 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.

Wrong! An alternate way is to well-order the rationals so as to have
only one non-successor, which has been done. See below:

Consider this well-ordering of the rationals:
Each rational, n/d, is represented by the quotient of
an integer numerator, n,
and a natural number denominator, d,
with no common integer divisors greater than 1,
then define a new ordering on the rationals so that
n1/d1 > n2/d2 if and only if
either | n1 | + d1 < | n2 | + d2
or both | n1 | + d1 = | n2 | + d2 and n1 < n2.

Then the set of all rationals reordered as above described is
order-isomorphic to the naturally well-ordered set of naturals,
producing a natural bijection between |Q and |N.
>

>
> Proof (2): The putative enumeration is a super task

It is a finite task completed above by gining a finite definition of the
new well-ordering relation, since, given any non-empty set of
rationals, there is a smallest one by that well-ordering.

WM is so hot on having finite definitions,
so why does he then ignore this one.

>
> Did you understand it? Or what is your counter argument?

We ask what is WM's couner argument to the well-ordering of |Q above,
but knowing that, since it proves him wrong, WM will only ignore it.
> >
> > 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.

> The mistake lies only in the false interpretation of "set" as an actually
> existing object.

Nothing in mathematics is "actually existing" in any physical sense.
Not even numbers. They are merly ways of thinking.

And WM's ways differ from most.

"As far as the laws of mathematics refer to
reality, they are not certain; and as
far as they are certain, they do not refer
to reality. It seems to me that complete
clearness as to this state of things first
became common property through that new departure
in mathematics which is known by the name of
mathematical logic or Axiomatics.¹ The progress
achieved by axiomatics consists in its having
neatly separated the logical-formal from its
objective or intuitive content; according to
axiomatics the logical-formal alone forms the
subject-matter of mathematics, which is not
concerned with the intuitive or other content
associated with the logical-formal. . . .
[On this view it is clear that] mathematics
as such cannot predicate anything about
perceptual objects or real objects. In
axiomatic geometry the words point,¹ straight
line,¹ etc., stand only for empty
conceptual schemata."

Albert Einstein
--

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