On Friday, 1 August 2014 19:34:39 UTC+2, Martin Shobe wrote:
> > The number of unit intervals, each one containing infinitely many rationals without index =< n, increases infinitely, i.e., beyond any upper bound.
> > 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. > > > > > Of course a matheologian will brush this aside by the standard blether "cardinals are not continuous". > > > > 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. > > > > > 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. This, however, is a very naive way of thinking that infinite sets can be exhausted like finite sets. > > 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.