Why can no one in sci.math understand my simple point?
in [Logic]
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From: Tim Little on 28 Jun 2010 05:35 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. - Tim
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. |