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Topic: ? 533 Proof
Replies: 46   Last Post: Aug 4, 2014 8:39 PM

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Virgil

Posts: 10,821
Registered: 6/8/11
Re: ? 533 Proof
Posted: Aug 1, 2014 6:17 PM
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In article <6bdfa706-a6d9-4d6f-8618-697046e5ad43@googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:

> On Friday, 1 August 2014 19:34:39 UTC+2, Martin Shobe wrote:

> > This is not evidence that the rationals cannot be enumerated by the
> > naturals.

>
> It is, because only natural numbers can index. But for every natural numbe I
> can show that it is not sufficient. Therefore you must believe in something
> unmathematical. To beleiev this requires a cerebral defect that most
> mathematicians do not have acquired.


And n view of the following, proper mathematicians also have not
acquired it and manage to prove it false.

Here is a straightforward way to construct a well-ordering of the
positive rationals. Write each one as p/q where naturals p and q have no
common factor (other than 1). Order them by ascending value of (p+q),
then within each set of p+q values, order by ascending p.

So you get the well ordering:
1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1...,
with each positive rational having a naturally numbered position in that
well -ordering

Anyone other than WM can also well-order the whole set of rationals by
putting zero at the start and interleaving each negative after its
corresponding positive.

> >
> > > Of course a matheologian will brush this aside by the standard blether
> > > "cardinals are not continuous".


Actual mathematicians have much better arguments than that! See Above!
> >
> >
> > I have no idea what a matheolgian would do, but a mathematician wouldn't
> > brush it aside that way. They would point out that "the rationals cannot
> > be enumerated by the naturals" doesn't follow from "The number of unit
> > intervals, each one containing infinitely many rationals without index
> > =< n, increases infinitely".

>
> It follows from the proof that every natural numbers fails. Enough for a
> mathematician.


Not enough for any mathematician who can read and uderstand the above
proof that both Q+ and Q can be well-ordered order-isomprphically to the
natural ordering of N.
> >
> >
> >

> > > But everybody with a critical intellect will ask *why* he should believe
> > > this.

> >
> >
> >
> > And everybody with an ounce of mathematical ability will notice that
> > it's because we can prove it.

>
> For that "proof" you have to assume that every is tantamount with all.


But mathematicians don't have to assume any such thing to prove the
reevant well-orderings.
Mathematicians are easily satisfied by the proof above that both Q+ and
Q can be ordered order-isomorphically to the standard ordering of N

>This, however, is a very naive way of thinking that infinite sets can
> be exhausted like finite sets.


An finitely defined order relation on a set applies to all members
simultaneously, even if there are more than finitely many of them.


> >
> > Back to the Ad Hominems again.
> >

> Unfortunately a thought is not independent of the human who thinks it. If a
> proof shows that every natural fails but you can "prove" the contrary, based
> on the false assumption that infinite sets can be exhausted like finite sets,
> then only a defect can prevent you to recognize this obvious error.


The above proof via well-orderings relies only on universal properties
of well-orerings which hold for finite sets and for both potentially
infinite and actually infinite sets.

So by that above proof there IS a bijection between N and Q which holds
even in WM's worthless world of WMytheology.
--
Virgil
"Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)



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