In article <5400048f-9123-4823-bf4a-06769640ef87@q12g2000yqj.googlegroups.com>, Transfer Principle <lwalke3@lausd.net> wrote:
> On Jun 17, 3:16 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote: > > Virgil <Vir...@home.esc> writes: > > > In article > > > <995d761a-f70f-4bca-b961-8db8e1663...@d37g2000yqm.googlegroups.com>, > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > >> On 15 Jun., 22:24, Virgil <Vir...@home.esc> wrote: > > >> > Note that it is possible to have an uncomputable number whose > > >> > decimal expansion has infinitely many known places, so long as it > > >> > has at least one unknown place. > > >> That is mathematically wrong. > > > It may not match every definition of 'uncomputable', but otherwise it is > > > right. > > It doesn't really match any definition of 'computable'. > > It's a rare day indeed when Aatu and WM actually agree! > > They both agree that Virgil's definition of "uncomputable" > is wrong. Let's see what's at stake here. > > We all agree that Chaitin's omega is uncomputable, so let > us define another real number as follows. > > Suppose we regard Chaitin's omega in let the number x equal > 1 if there exists a natural number N such that for all > natural numbers n>N, we have that a majority of the first n > digits of omega are 1's, and let x equal 0 othewise. So now > we ask, is x a computable real? > > By Virgil's definition, it isn't, since there's no Turing > machine that can compute whether x=0 or x=1. > > But by Aatu and WM, it is computable, since after all, x > must be either 0 (which is computable), or 1 (which is also > computable), so in either case, x is computable. > > We recall that in classical logic, we know that from P->R > and Q->R, we conclude (PvQ)->R. So we have: > > P <-> "x=0" > Q <-> "x=1" > R <-> "x is computable" > > So P->R is "if x=0, then x is computable" (which is true, > since 0 is computable). > > So Q->R is "if x=1, then x is computable" (which is true, > since 1 is computable). > > So we conclude (PvQ)->R, which is "if x=0 or x=1, then x > is computable." And we know that x must be 0 or 1. Thus, > by Modus Ponens, x is computable. QED > > So which definition of computable, Virgil's or Aatu-WM's, > should we use? I'm not sure which of the two definitions > is more prevalent.
We could distinguish between them by using "uncomputable" for one of them and "noncomputable" for the other.
Since so many seem to think that mine should NOT be called "uncomputable, lets call mine "noncomputable".
