Why can no one in sci.math understand my simple point?
in [Logic]
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From: Virgil on 17 Jun 2010 16:06 In article <7fa0f961-d798-4f6c-9b92-fba5a603db47(a)e5g2000yqn.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 17 Jun., 05:26, Tim Little <t...(a)little-possums.net> wrote: > > On 2010-06-15, Peter Webb <webbfam...(a)DIESPAMDIEoptusnet.com.au> wrote: > > > > > No. You cannot form a list of all computable Reals. If you could do > > > this, then you could use a diagonal argument to construct a > > > computable Real not in the list. > > > > You can form a list of all computable reals (in the sense of > > mathematical existence). �However, such a list is not itself > > computable. > > That does not matter. > There exists a list containing all computable reals in all possible > languages. > Therefore the set of reals that can serve as doiagonals of a Cantor > list is countable. > > Regards, WM Note, "all possible languages" is much too ambiguous to define a valid set of objects, so there is no such "set" of objects to list.
From: MoeBlee on 17 Jun 2010 16:14 On Jun 17, 2:00 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 17 Jun., 19:36, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > > Re: Jun 17, 12:23 pm, MoeBlee <jazzm...(a)hotmail.com>e: > > > P.S. > > > If the discussion about the formal theory ZFC, then I don't know what > > lack of trichotomy you're referring to. > > > Of course, many relations fail to satisfy trichotomy. But the standard > > strictly less than ordering on the reals satisfies trichotomy: > > > If x and y are reals, then exactly one of these holds: > > > x <_r y > > y <_r x > > x=y > > > where '<_r' stands for the standard stricly less than relation on the > > set of real numbers. > > That means, even undefinable real numbers are in trichotomy with each > other and in particular with definable real numbers? You said trichotomy is not important in this discussion. Anyway, if 'definable real number' is given a definition in ZFC and every 'definable real number' is a real number, then of course, trichotomy holds. Listen, you don't need to waste our time. If I say trichotomy holds among all real numbers than I mean just what I said. > How can that be? There is no way to name an undefinable real (because > there are only countably many names). And it is impossible to define a > real number by an infinite sequence, because onl finite sets and > sequences can be defined by listing the elements or terms. We prove a theorem that the ordering satisfies trichotomy. All the rest of your perceived difficulties with this have no bearing on that. Meanwhile, if you think we prove two contradictory theorems, then just state the exact P such that you think P and ~P are proven in ZFC. > Have you ever tried to put an undefinable real number in order with > other, definable real numbers? Whatever I have or have not tried has no bearing on what is or is not a theorem of ZFC. > > Or to be pedantic: > > > <x y> e <_r > > <y x> e <_r > > x=y > > > And there is no ordering on the cardinals, > > OK. Let us stay with the reals. Say you have two undefinable reals > between 0 an 1. How can you manage to find out which one is less than > the other? To say that trichotomy is a theorem is not to say also that we have a way to find out anything at all. We have found out that there is a proof of the trichotomy of the reals. That's all I claim in this immediate regard. Look, please don't waste our time. I said clearly that it is not at issue with me that you may find many contradictions between ZFC and your own mathematical notions. Such contradictions that you, in particular, find, are not of particular interest to me at this time. On the other hand, you say you do not care about formal theories. So there is little for us to discuss. On the other hand, when you do talk about ZFC or about common arguments that are also formalized in ZFC, then I may wish to comment on your remarks. If you have a contradiction between ZFC and your mathematics, then you don't have to convince me since I well recognize that you may find such contradictions, but if you claim something "wrong" in ZFC itself, then please produce an actual contradiction IN ZFC or else we are just back to the fact that you find contradictions between ZFC and your own mathematics. MoeBlee
From: Virgil on 17 Jun 2010 16:31 In article <627bf325-9dbb-4967-a878-7a0bfd8a4677(a)s9g2000yqd.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 17 Jun., 09:54, Virgil <Vir...(a)home.esc> wrote: > > In article <4c19cd2c$0$316$afc38...(a)news.optusnet.com.au>, > > �"Peter Webb" <webbfam...(a)DIESPAMDIEoptusnet.com.au> wrote: > > > > > Cantor's proof applied to computable numbers proves you cannot form a > > > computable list of computable numbers. Cantor's proof applied to Reals > > > proves you cannot form a computable list of Reals. > > Cantors proof is nonsense from the beginning, because a real number > can never be defined by an infinite sequence alone. Maybe not to WM, but he is is not GOD to determine what is and what is not true. It is clear that what he thinks is mostly irrelevant here and in mathematics in general. > A definition > defines something, but an infinite sequence does not define a number > before the last digit is known. Only such little minds as WM has can argue that an infinite sequence has a last member. > > > > To be correct, there is no computable list of ALL of the computable > > numbers, even though the set of computable numbers is e countable, but > > there are lots of possible computable lists of computable numbers. > > And there are lots of lists of more than all computable numbers, > namely lists of all finite expressions. Which statement concedes the existence of infinite lists. Which justifies the "Cantor diagonal" argument.
From: Virgil on 17 Jun 2010 18:28 In article <cd66b418-fe7b-472a-8fbe-527173e0aaf4(a)c33g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 17 Jun., 19:36, MoeBlee <jazzm...(a)hotmail.com> wrote: > > Re: Jun 17, 12:23�pm, MoeBlee <jazzm...(a)hotmail.com>e: > > > > P.S. > > > > If the discussion about the formal theory ZFC, then I don't know what > > lack of trichotomy you're referring to. > > > > Of course, many relations fail to satisfy trichotomy. But the standard > > strictly less than ordering on the reals satisfies trichotomy: > > > > If x and y are reals, then exactly one of these holds: > > > > x <_r y > > y <_r x > > x=y > > > > where '<_r' stands for the standard stricly less than relation on the > > set of real numbers. > > That means, even undefinable real numbers are in trichotomy with each > other and in particular with definable real numbers? But it does not mean that we can ALWAYS determine which of two reals is the larger. > > OK. Let us stay with the reals. Say you have two undefinable reals > between 0 an 1. How can you manage to find out which one is less than > the other? Perhaps you don't. That one of a limited number of options is the correct one does not necessarily mean that which one is correct can be determined. In every sufficiently complex system, including those sufficiently complex to support standard arithmetic, there are statements whose truth cannot be determined within that system.
From: Transfer Principle on 17 Jun 2010 20:35
On Jun 17, 3:16 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Virgil <Vir...(a)home.esc> writes: > > In article > > <995d761a-f70f-4bca-b961-8db8e1663...(a)d37g2000yqm.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > >> On 15 Jun., 22:24, Virgil <Vir...(a)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. |