In article <email@example.com>, firstname.lastname@example.org wrote:
> 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.
That "proof" is invalid unless you can also prove that the limit/union of an increasing sequence of finite sets must be finite, which is not true outside of your WMytheology.
> You cannot > disprove it, because you cannot find any element that stays outside, can you?
One can find counterexamples. For example, let S_n be the normally ordered set of integers between -n and n inclusive. Each S_n, being finite, is well-ordered under its standard ordering, but their union, not being finite, turns out also not to be well-ordered.
Thus WM's assumption that the union of an increasing sequence of well-ordered sets must be well-ordered is seen to be false, at least false outside of WM's wild weird world of WMytheology, just as his claim that the union of an increasing sequence of finite set must be finite. > > > 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.
Finite ordered sets are necessarily well-ordered Non-finite ordered sets are not necessarily well ordered.
Since any countably infinite ordered set can be expressed as the limit/union of a sequence of finite, thus well-ordered, ordered sets, WM is in effect claiming that every countably infinite set is well-ordered.
The evidence of a lots of such sets which are countably infinite but not well-ordered sows that his assumption that infinite unions of well-ordered sets need not be well-ordered, at least not outside the wild weird world of WMytheology.
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. > > Regards, WM --