Virgil
Posts:
8,833
Registered:
1/6/11


Re: Matheology � 413
Posted:
Dec 31, 2013 4:29 PM


In article <281133407e76400daa341c58de49b538@googlegroups.com>, mueckenh@rz.fhaugsburg.de wrote: > On Tuesday, 31 December 2013 16:37:31 UTC+1, wpih...@gmail.com wrote: > > On Tuesday, December 31, 2013 11:03:35 AM UTC4, muec...@rz.fhaugsburg.de > > wrote: > > > On Tuesday, 31 December 2013 15:22:09 UTC+1, wpih...@gmail.com wrote: > > > > On Tuesday, December 31, 2013 7:05:58 AM UTC4, > > > > muec...@rz.fhaugsburg.de wrote: > > > > > Irrational numbers have no decimal (or binary or whatever > > > > > integerpositivebase) expansion. > > > It is impossible for an infinite list of decimals to appear in > > > mathematical discourse, dialogue, or monologue other than as the finite > > > rule how to calculate *every* decimal at a finite place > > > > No you need a finite definition of a potentially infinite decimal > > > > sequence. > > > > This can define an irrational number. > > > Of course. > > Everything that you say cannot be done with > > digits can be done with finite definitions of potentially infinite > > sequences e.g. diagonalization. > > Indeed, it can easily be shown that given a potentially infinite sequence, > > L, > > of finite definitions of potentially infinite 0/1 sequences there is > > a finite definition of a potentially infinite 0/1 sequence that is not > > an element of L.
> You think it is impossible to list all finite definitions?
It is certainly impossible for WM to do it so long as he rejects the possiblity of actual infinteness.
> Then it is impossible to list all terminating rationals.
Every rational terminates in some base.
Except that a listing of ALL rationals has been done by any of a huge number of wellorderings of them
> It is easy to verify that the list of all terminating rationals is in > bijection with the list of all finite words. Note how WM often claims things are easy, but never manages ever to do any of them, just as WM often claims things are impossible to do when others have actually done them. 

