Why can no one in sci.math understand my simple point?
in [Logic]
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From: Virgil on 26 Jun 2010 15:30 In article <511dd6ef-6fe7-4664-bda0-082971c2f8db(a)z10g2000yqb.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 26 Jun., 00:32, "Mike Terry" > > > > Please let me know: Does the list consisting of A0, A1, A2, A3, ... > > > contain its antidiagonal or not? > > > > No, of course not. > > The set of all these antidiagonals A0, A1, A2, A3, ... constructed > according to my construction process and including the absent one is a > countable set. > > > > > > > That is not a problem. It shows however, that there are countable sets > > > that cannot be listed. > > > > How exactly does it show that there are countable sets that cannot be > > listed? > > > See the one above: The results of my construction process. > > > > > OK, but you've still not given any explanation why you think the > > antidiagonals of your process cannot be listed! (I'm genuinely baffled.) > > You said so yourself, few lines above. Saying something does not make it true. WM, for example, often says things which are not true. Since listable and countable are in all respects identical (logically equivalent properties), there cannot be any set provably countable and provably not listable.
From: Virgil on 26 Jun 2010 15:35 In article <8ced201d-fce9-447f-9fa9-b972f2014d8e(a)t5g2000prd.googlegroups.com>, Newberry <newberryxy(a)gmail.com> wrote: > On Jun 25, 11:22�pm, "Peter Webb" > <webbfam...(a)DIESPAMDIEoptusnet.com.au> wrote: > > "Virgil" <Vir...(a)home.esc> wrote in message > > > > news:Virgil-71EA75.23485925062010(a)bignews.usenetmonster.com... > > > > > In article > > > <b3413a4e-567b-4dfb-8037-21f14b826...(a)g1g2000prg.googlegroups.com>, > > > Newberry <newberr...(a)gmail.com> wrote: > > > > >> > > No. (3) is not true, as it is based on a false premise (that the > > >> > > computable > > >> > > Reals can be listed). > > > > > How is countability any different from listability for an infinite set? > > > > > Does not countability of an infinite �set S imply a surjections from N > > > to S? And then does not such a surjection imply a listing? > > > > It implies a listing must exist, but does not provide such a listing. > > Does such a listing have an anti-diagonal? Every list of reals, or of infinite binary sequences, has an antidiagonal, in fact, has at least as many antidiagonals as elements. > > > The computable Reals are countable, but you cannot form them into a list of > > all computable Reals (and nothing else) where each item on the list can be > > computed. > > > > In order to list a set, it has to be recursively enumerable. Being countable > > is not sufficient. In order explicitely to construct such a list, perhaps, but such lists may exist even when not explicitely constructable. And for any countable set, they do.
From: Virgil on 26 Jun 2010 15:40 In article <1a5a0da4-01d1-41c5-b64b-dcc53e5ce8f2(a)k39g2000yqb.googlegroups.com>, Charlie-Boo <shymathguy(a)gmail.com> wrote: > On Jun 25, 3:25�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <adaa005c-c180-4ce0-8d2c-81da912c6...(a)w12g2000yqj.googlegroups.com>, > > > > > > > > > > > > �Charlie-Boo <shymath...(a)gmail.com> wrote: > > > On Jun 24, 8:49 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > Charlie-Boo <shymath...(a)gmail.com> writes: > > > > > On Jun 15, 2:15 am, "Peter Webb" > > > > > > >> No. You cannot form a list of all computable Reals. > > > > > > > Of course you can - it's just the list of Turing Machines. > > > > > > No, it's not. > > > > > I asked for a counterexample, to no avail. �Don't you think you should > > > substantiate your statement or retract it? > > > > > Each Turing Machine represents some computable real (all computable > > > reals are included) and you can list those Turing Machines. �The > > > Turing Machine represents it as well as any other system of > > > representation. > > > > Aren't there Turing machines that don't represent any real at all? > > No. In general terms, every TM computes some result from its input. > Agree? Then if we start with an empty tape, it represents a > constant. How we map its execution history determines which real > number that represents. There is generally a way to indicate which > actions constitute output, which is needed when we use an infinite > representation such as its real number binary expansion. > > If the executon history can be finite, we can map all finite strings > into real numbers, as well as the infinite ones as described above. How does one associate TMs which are nonterminating but cyclic and whose cycle depends on the input tape?
From: Virgil on 26 Jun 2010 15:45 In article <b5f0d60a-f8d6-4e40-a180-a9dbbfd28b05(a)u26g2000yqu.googlegroups.com>, Charlie-Boo <shymathguy(a)gmail.com> wrote: > It is trivial to calculate in binary (using only 0 and 1) and output > any desired string surrounded by 2 whenever we want, then immediately > erase one of the 2's. Thus every real number will be defined by some > TM. How does one calculate one of those many reals that one cannot calculate, the inaccessible ones? Note that most reals are inaccessible, uncountably many of them, with only countably many being accessible.
From: Mike Terry on 26 Jun 2010 17:14
"WM" <mueckenh(a)rz.fh-augsburg.de> wrote in message news:511dd6ef-6fe7-4664-bda0-082971c2f8db(a)z10g2000yqb.googlegroups.com... > On 26 Jun., 00:32, "Mike Terry" > > > > Please let me know: Does the list consisting of A0, A1, A2, A3, ... > > > contain its antidiagonal or not? > > > > No, of course not. > > The set of all these antidiagonals A0, A1, A2, A3, ... constructed > according to my construction process and including the absent one is a > countable set. Yes, I've agreed that several times. > > > > > > > That is not a problem. It shows however, that there are countable sets > > > that cannot be listed. > > > > How exactly does it show that there are countable sets that cannot be > > listed? > > > See the one above: The results of my construction process. No. The above shows that you've constructed a countable sequence of numbers (A0, A1, A2, A3,...) which CAN be listed, namely consider the list (A0, A1, A2, A3,...). Read carefully! > > > > > OK, but you've still not given any explanation why you think the > > antidiagonals of your process cannot be listed! (I'm genuinely baffled.) > > You said so yourself, few lines above. Rubbish. I have never said anything other than that (A0, A1, A2, ...) is a countable sequence of reals that CAN be listed. Go on - have one more go, then I'll give up... Mike. > > Regards, WM |