Why can no one in sci.math understand my simple point?
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From: Jesse F. Hughes on 28 Jun 2010 08:48 Tim Little <tim(a)little-possums.net> writes: > On 2010-06-27, Jesse F. Hughes <jesse(a)phiwumbda.org> wrote: >> I haven't seen his definition, but he seems to suggest that the >> machine that repeatedly overwrites a single square with 0 or 1 >> computes a real number. Is that consistent with his definition? > > Yes, apparently it represents the real number 0, as all machines do > that never produce a tape state having at least two "2" symbols. > > As I recall, the mapping goes something like this: the successive > states in which there exist at least two "2" symbols on the tape > define a sequence of binary strings between the leftmost pair of "2"s. > Then there is a mapping from sequences of binary strings to reals. > > I don't think the latter mapping was explicitly given, but there are > plenty of suitable options. Okay, thanks for the correction. Pretty odd notion of computing a real number, but I'll take your word for it that it's equivalent to the usual notion. -- "[I]f I could go back, [...] I would tell myself not to step into a position where the fate of the entire world could rest in my hands. I would [avoid this] path to a nightmarish and surreal world, a topsy-turvy world, where everything changes." -- James S. Harris cannot escape his destiny.
From: Daryl McCullough on 28 Jun 2010 12:40 Tim Little says... > >On 2010-06-28, Virgil <Virgil(a)home.esc> wrote: >> It does not require that any element in the listing be known, but >> correctly tells what to do for any listing > >I think that is even a bit too informal for Peter. The phrase "tells >what to do" is superfluous, all that is mathematically required is >that existence of an antidiagonal sequence for each list is proven. >He's going to latch onto "tells what to do" and think that it means >that there is an algorithm for everything involved. > >Witness his confusion over the example I defined of a list where each >entry was computable but the list itself (and its antidiagonal) was >not. He didn't dispute that the list *existed*, but considered it >cheating because he couldn't use the definition to extract actual >digits of the antidiagonal - it didn't "tell him what to do" in his >own special sense. I never saw a response from Peter on my post about Turing machine computability relative to an oracle. You can imagine a Turing machine tape (the oracle) that lists codes for computable functions. Using the oracle, you can diagonalize to get a new function that is not listed by the oracle. This new function is computable *relative* to the oracle, but may not be computable *without* the oracle. -- Daryl McCullough Ithaca, NY
From: Virgil on 28 Jun 2010 13:46 In article <d480f2f1-3097-4324-b351-fb796ceb418b(a)y4g2000yqy.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 28 Jun., 11:07, Tim Little <t...(a)little-possums.net> wrote: > > On 2010-06-28, Peter Webb <webbfam...(a)DIESPAMDIEoptusnet.com.au> wrote: > > > > > This occurs in step 4. You state we can "identify an element d of S > > > that is on the diagonal". Unless the list is explicitly defined, you > > > can't "identify" the digit in position x. > > > > See Owen? �I told you he wouldn't be able to ignore the informal fluff > > phrase "we can identify", and so fails to interpret it correctly. > > > > Mathematically, it means nothing more than "there exists". > > And mathematically "there exists" means nothing. Maybe that is the case in WM's world, but it is quite different in everyone else's world.
From: Virgil on 28 Jun 2010 13:52 In article <slrni2gp3j.jrj.tim(a)soprano.little-possums.net>, Tim Little <tim(a)little-possums.net> wrote: > On 2010-06-28, Owen Jacobson <angrybaldguy(a)gmail.com> wrote: > > So, here is an informal presentation of Cantor's diagonal argument > > that avoids the word "list" (as well as a few other common verbal > > shortcuts): > > > > 1. Let S be the set {0, 1}^N. > > 2. For any function L from N to S, we can identify an element of S not > > in the image of L. > > Peter is incapable of separating the usual informal phrase "we can..." > from his fixed idea of "there exists a finite algorithm that can...". Note that the "anti-diagonal algorithm" which produces the anti-diagonal from a list of binary sequences is itself quite clearly finite, even though it acts on something which is not.
From: Virgil on 28 Jun 2010 13:55
In article <slrni2gr79.jrj.tim(a)soprano.little-possums.net>, Tim Little <tim(a)little-possums.net> wrote: > On 2010-06-28, Virgil <Virgil(a)home.esc> wrote: > > It does not require that any element in the listing be known, but > > correctly tells what to do for any listing > > I think that is even a bit too informal for Peter. The phrase "tells > what to do" is superfluous, all that is mathematically required is > that existence of an antidiagonal sequence for each list is proven. > He's going to latch onto "tells what to do" and think that it means > that there is an algorithm for everything involved. There are certainly simple algorithms for finding up to countably many antidiagonals to any given list of reals or of binary sequences. |