Why can no one in sci.math understand my simple point?
in [Math]
|
Prev: power of a matrix limit A^n
Next: best way of testing Dirac's new radioactivities additive creation Chapt 14 #163; ATOM TOTALITY
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.
From: Virgil on 16 Jun 2010 16:46
In article <7a3e3a5a-2202-4cbb-9556-e3766ee5a745(a)k39g2000yqd.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 16 Jun., 17:55, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > WM <mueck...(a)rz.fh-augsburg.de> writes: > > > It is very probable that every line has many different meaning, but no > > > line has uncounatbly many meanings. > > > > Every line has an indefinite and indeterminate number of possible > > meanings. It makes no sense to speak of the cardinality of the totality > > of all possible meanings of a string of symbols. > > It is fact that all possible meanings must be defined by finite > definitions. Therefore the meanings are countable. Therefore it makes > sense to call the set of all meanings countable. In proper set theory, one must be able to determine of an object whether or not it is a member of a set, but since those vague "meanings" of a string of symbols are so indeterminate, one cannot determine whether a "meaning" does belong to a "symbol" so WM's "sets" are not well enough defined to be sets in any proper set theory. > > In mathematics we can calculate or estimate things even if not all can > be named. But set membership must be less ambiguous than WM's. > > > > > Therefore the list contains only countably many finite definitions. Except that it is not at all clear what has been listed. > > > > The list contains just random, meaningless strings. Whenever we instill, > > with mathematical precision, these strings with some definite meaning, > > so that they become definitions of reals, we find there are definitions > > not included in the list. There is an absolute notion of computability > > in logic. > > If this notion yields numbers only that are in trichotomy with each > other, then the notion is acceptable. If this notion yields numbers > like you gave examples for (IIRC) like n = (1 if Obama gets a second > term and 0 otherwise) or so, then this notion together with your logic > should be put into the trash can. Only for a while. > > > There is no absolute notion of definability. > > We need not an absolute notion if we know that all possible > definitions of the notion of definability belong to a countable set. We cannot possibly know that. > > It is enough to prove by estimation that set theory is wrong. Compare > the famous irrationality proofs and transcendence proofs of number > theory. We need not calculate its deviation from truth to the fifth > digit. It is enough to see that ZFC is wrong unless there is a natural > between 0 and 1. But what everyone else sees is that ZFC is right unless there is a natural between 0 and 1. |