Virgil
Posts:
10,821
Registered:
6/8/11


Re: ? 533 Proof
Posted:
Aug 1, 2014 6:17 PM


In article <6bdfa706a6d94d6f8618697046e5ad43@googlegroups.com>, mueckenh@rz.fhaugsburg.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 wellordering 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 wellorder 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 wellordered orderisomprphically 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 wellorderings. Mathematicians are easily satisfied by the proof above that both Q+ and Q can be ordered orderisomorphically 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 wellorderings relies only on universal properties of wellorerings 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)

