On Wednesday, 12 June 2013 17:59:40 UTC+2, Zeit Geist wrote: > > Every configuration can be well-ordered by size. This implies there is no q that stays outside the well order.
> 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?
> Any finite set of real numbers can be well-ordered according to magnitude. Infinite ones, not necessarily.
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.
I claim: Every set of natural numbers can be put in every desired permutation. Among them there is the permutation that orders Q by size.