Re: Cantor Diagonal Proof

Four fairly random comments:
   (a) There have been several references in this thread to the fact that
we can't store the whole of an infinite sequence and similar.  Perhaps
worth noting that there is a fundamental difference between [pure] maths
and CS in this area.  In maths, there are practical problems in numerical
analysis, in convergence, and the like, but worrying about storage is /so/
20thC.  If you have a list of numbers, then you can let a[i,j] be the j-th
decimal digit of the i-th number, and it just /is/.  Similarly the number
C whose n-th digit is an appropriate tweak of a[n,n] just /is/, and it is
easily seen to be different from any number in your list.  There is no
question about waiting for the computation of C to finish or of having to
store it anywhere.
   (b) Turing died in 1954, and published his best known paper in 1937, ie
before WW2, before there were any electronic computers, before there was
any recognisable CS, and /long/ before there were any undergrad CS courses.
He is, of course, a very important pioneer of the subject, but both maths
and CS have moved on a long way since 1937, and everything he writes about
computability needs to be treated with caution.
   (c) I don't know whether I'm the only person to recognise the convergence,
but both our resident cranks remind me of a Greek tragedy:
     Petrence:  Here is a spiffing new idea!
         Chorus:  It's neither new nor spiffing.
     Lawer:  Anyone who understands can see that I'm right.
         Chorus:  But the first mistake is [whatever].
     Petrence:  You don't understand, here is my proof.
         Chorus:  It's wrong because [whatever].
     Lawer:  No-one can point to a mistake.
         Chorus:  Yes, we can, and have!
     Petrence:  You don't understand, here is my proof.
         Chorus:  It's wrong because [whatever].
     [Repeat ad inf or ad nauseam, take your pick]
I for one am happy to cut a newcomer with a problem or an idea some slack,
but not to be the third spear-carrier in the chorus.  Clearly some other
knowledgeable posters feel the same way.  Others ... less so.
   (d) Despite RichardH's comments, I think the jury is still just about
out on Wij.  His root problem is that he doesn't accept the Archimedean
[or Eudoxus] axiom [essentially that there are no infinitesimal real
numbers].  Some of the weirder things he says are more-or-less correct in
some non-standard systems, but he refuses to learn about NS analysis or
about the surreals, and is [unsurprisingly] incapable of putting his theory
onto a firm, axiomatic basis.  Likewise with Peter's ideas about "geometric
points".
--
Andy Walker, Nottingham.
    Andy's music pages: www.cuboid.me.uk/andy/Music
    Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Chwatal