In article <firstname.lastname@example.org>, email@example.com 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)