On Wednesday, 12 June 2013 22:59:24 UTC+2, Zeit Geist wrote:
> > > Every finite one can, but not the entire set of Q.
> > I prove that all elements of Q can be well-ordered by size. You cannot disprove it, because you cannot find any element that stays outside, can you? > > > > You proved nothing!
Your nded unfouopinion is not of interest in this connection. > > You proposed an algorithm, > > But never showed that it conclusively does what you say.
You can take whatever algorith you like to well-order by size any finite set of rationals. Since every rational number q_i belongs to a finite set of i rationals, this proves that all rationals can be ordered by size - if there is something like all rationals. > > > > > > > Any finite set of real numbers can be well- according to magnitude. Infinite ones, not necessarily.
I stated that all rationals that belong to FIS of the enumeration can be well-ordered by size. >
> > > > > Either find a q that stays outside of the well-order by size. If such a q exists and is enumerated as q_n, then there is a first natural number n that cannot be put in a permutation such that all q's are in order by size. If such a q exists and cannot be enumerated, then countability is nonsense.
> Sure thing. > > If the enumeration has f(j) = 1/2, > > what is f(j+1), f(j+2) and f(j+3)?
That depends on the enumeration you choose at the outset. Take Cantor's original one. > > > > If f keeps the order we must have > > > > | f(m) | <= | f(m+1) |
f does not keep the order. Always when you extend the set of rationals, you order them by size. This is possible for every finite set, i.e., for every FIS of the enumerated set. > > > > And no rational r, such that the magnitude > > Of r is in between that of f(m) and f(m+1).
You are lacking understanding.
Take the first i rationals, order them by size. Add the rational with number i+1put them in the correct order by size with the others. And so on without ever arriving at an infinite i and without leaving out any rational q_i or q_i+1 or any desired rational.