In article <firstname.lastname@example.org>, email@example.com wrote:
> On Sunday, 3 August 2014 16:38:53 UTC+2, V I R G I L wrote: > > > > > > What |N does that no finite initial segment of |N does is contain ALL > > finite initial segments of |N as proper subsets, so |N does things that > > no finite initial segment of |N can do. > > And for just all these finite initial segments I have shown that they leave > infinitely many, in fact increasingly many, rationals without index. Whereas > you show your enumerations only for some 10 naturals. Feeble.
WM lies! Actually what I HAVE shown is at least two slightly different bijections between |N and |Q, each by ordering |Q order-isomorphically to the standard ordering of |N by means of a general definition of an order-relations on |Q which well-order |Q order-isomorphically to |N and thus bijects |N and |Q.
And each of those well-orderings of Q disproves WM's false claim that no such bijecting well-orderings are possible.
Here is a straightforward way to construct a well-ordering of the positive rationals. Write each one as p/q where naturals p and q have no common factor (other than 1). Order them by ascending value of (p+q), then within each set of p+q values, order by ascending p.
So everyone but WM gets t the well-ordering of Q+ starting with: 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, ...
Anyone other than WM can then well-order the whole set of rationals, Q, by putting zero first then interleaving each negative after its corresponding positive in the above ordering.
So everyone but WM understands the well-ordering of Q starting with: 0/1, 1/1, -1/1, 1/2, -1/2, 2/1,-2/1, 1/3, -1/3, 3/1, -3.1, 1/4, -1/4, 2/3, -2/3, 3/2, -3/2, 4/1,-4/1, ...
And everyone but WM now understands that WM is hereby proved totally and irrevocably wrong! And that WM's worthless world of WMytheology is anti-mathemtaics. -- Virgil "Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)