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