Why can no one in sci.math understand my simple point?
in [Logic]
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From: Virgil on 16 Jun 2010 16:04 In article <9df240be-eaec-4d46-bd74-42868f4970ec(a)g19g2000yqc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 16 Jun., 02:39, "Peter Webb" <webbfam...(a)DIESPAMDIEoptusnet.com.au> > wrote: > > > Nevertheless your "definition" belongs to a countable set, hence it is > > > no example to save Cantors "proof". > > > > > Either all entries of the lines of the list are defined and the > > > diagonal is defined (in the same language) too. > > > > Yes. If you provide a list of Reals, then the diagonal is computable and > > does not appear on the list. > > Delicious. Cantor shows that the countable set of computable reals is > uncountable. That would require that one can have a list of all and only the computable numbers which is already known to be impossible. So WM is wrong again, as usual.
From: Virgil on 16 Jun 2010 16:06 In article <bf42a4e2-9f0f-42a4-8bfe-2e884b633982(a)c33g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > But all that is not of interest for the present problem: All > definable, computable, and somehow identifiable numbers and forms of > numbers are within a countable set. Once WM concedes any countably infinite set, as he does above, he has lost the battle.
From: Virgil on 16 Jun 2010 16:24 In article <a3e6534b-2b79-447d-aa6b-da536824e31c(a)x27g2000yqb.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 16 Jun., 11:48, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > "WM" <mueck...(a)rz.fh-augsburg.de> wrote ... > > > > > > > > > By induction we prove: There is no initial segment of the (ANTI)diagonal > > > that is not as a line in the list. > > > > Right, therefore the anti-diagonal does not contain any pattern of digits > > that are not computable. > > > Sorry, you misquoted me. I wrote: > By induction we prove: There is no initial segment of the diagonal > that is not as a line in the list. And there is no part of the > diagonal that is not in one single line of the list. > But I have to excuse because I wrote somewhat unclear. > > The meaning is: > 1) Every initial segment of the decimal expansion of pi is in at least > one line of your list > 3. > 3.1 > 3.14 > 3.141 > ... > What we can finde in the diagonal (not the anti-diagonal), namely > 3.141 and so on, exactly that can be found in one line. This is > obvious by construction of the list. > > 2) Every part of the diagonal is in at least one line. That means, > every part is in one single line, or there are parts that are in > different lines but not in one and the same. > > The latter proposition can be excluded. How is the latter excluded? Consider the sequence f(n) = trunc(pi*10^n)/10^n While there are some successive lines which are equal (where the decimal expansion of pi has 0 digits), for every n there is an m, with n < m, such that f(n) < f(m) < pi. > If there are more than one > lines that contain parts of pi, then it can be proved, be induction, > that two of them contain the same as one of them. This can be extended > to three lines and four lines and so on for every n lines. False for my example above. > > Hence we prove that all of pi, that is contained in at least one of > the finite lines of your list, is contained in one single line. False for my example above. And equally false for f(n) = trunc(x*10^n)/10^n with any real x which has a non-terminating decimal expansion.
From: WM on 16 Jun 2010 16:34 On 16 Jun., 21:40, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 16, 2:21 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > I don't have a thousand lifetimes to wait for you to show a > > formula P such that both P and ~P are derivable in ZFC from the above > > definition. > > CORRECTION: I should have said: > > [...] to show, for some formula P, a proof in ZFC (with said > definition included) of P and a proof in ZFC (with said definition > included)of ~P. Look simply at the results. If a theory says that there is an uncountable set of real numbers such each number can be identified as a computable or definable or constructable one, or in other ways, then this theory is provably wrong. Reason: Cantor either proves that a countable set is uncountable or that a constructible/computable/definable number is not constructible/ computable/definable. What the theory internally may be able to prove or not to prove is, at least for my person, completely uninteresting. Regards, WM
From: Virgil on 16 Jun 2010 16:37
In article <7399a634-9ad2-4c73-9c57-c26bf99a60cd(a)d8g2000yqf.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > This list is a list of everything. Then it must list itself. |