Cantor Confusion
in [Math]
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From: Ralf Bader on 17 May 2007 16:50 WM wrote: > There are different notions (for instance weakly normal numbers and > absolutely normal numbers). Oh yes. There are almost always different notions in mathematics. > Of course normal numbers can be > constructed, one of the simplest cases is the rational number > 0.12012012... with respect to base 3, but as there must be included > also normally distributed frequencies of 10^100-tuples and larger > tuples most normal numbers cannot be constructed. Congratulations! This sentence may very well be the most stupid nonsense ever written about normal numbers in all of world history.
From: WM on 17 May 2007 17:01 On 17 Mai, 13:21, William Hughes <wpihug...(a)hotmail.com> wrote: > On May 16, 3:55 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 14 Mai, 22:09, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > On May 14, 3:17 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > Which of the following statements is not true > > > > Every element of R is different from p. > > > > Every node of p is in some element of R > > > Back to the facts: > > Find a path p and a set of paths R such that > p is not in R, but every node in p is in some element of R. > > > > > 1) No element r =/= p of R is required to cover any node of p. (I can > > show of every such element r =/= p which you may name that it is not > > required.) > > So What? That every element of R was necessary was not required > and I never claimed it was true. Give a set of elements which are necessary. > > > 2) No element of R is sufficient to cove all nodes of p. > > So What? That no one element of R is sufficient was not required > and indeed I have noted it is impossible. Give a path or a set of paths which are necessary and sufficient to cover p. > > > Hence your set R is not suitable to demonstrate your claims. > > Does not follow. Which of these statements is not true. > > Every element of R is different from p. That is correct, but uninteresting. > > Every node of p is in some element of R That is wrong if p is different from every element of R. Regards, WM
From: Virgil on 17 May 2007 17:15 In article <1179435683.724641.326740(a)p77g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 17 Mai, 13:21, William Hughes <wpihug...(a)hotmail.com> wrote: > > On May 16, 3:55 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > On 14 Mai, 22:09, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > On May 14, 3:17 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > Which of the following statements is not true > > > > > > Every element of R is different from p. > > > > > > Every node of p is in some element of R > > > > > Back to the facts: > > > > Find a path p and a set of paths R such that > > p is not in R, but every node in p is in some element of R. > > > > > > > > > 1) No element r =/= p of R is required to cover any node of p. (I can > > > show of every such element r =/= p which you may name that it is not > > > required.) > > > > So What? That every element of R was necessary was not required > > and I never claimed it was true. > > Give a set of elements which are necessary. It is necessary that such a set be infinite, but no single member of it is necessary, as any infinite subset of such a set is still sufficient. There is no minimal set which is both necessary and sufficient, because there is no minimal infinite subset of an infinite set. WM cannot seem to sort these things out even with help/ > > > > > 2) No element of R is sufficient to cove all nodes of p. > > > > So What? That no one element of R is sufficient was not required > > and indeed I have noted it is impossible. > > Give a path or a set of paths which are necessary and sufficient to > cover p. Give a minimal infinite subset of N.Or any other actually infinite set. > > > > > Hence your set R is not suitable to demonstrate your claims. > > > > Does not follow. Which of these statements is not true. > > > > Every element of R is different from p. > > That is correct, but uninteresting. > > > > Every node of p is in some element of R > > That is wrong if p is different from every element of R. Not when we have actually infinite sets as ZF and NBG allow us, indeed require us, to have. That what we do in mathematics, WM cannot do in his MathUnRealism, only limits him. It does not limit us. Those who choose to wear blinders cannot force then on those who don't.
From: William Hughes on 17 May 2007 19:13 On May 17, 5:01 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 17 Mai, 13:21, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On May 16, 3:55 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > On 14 Mai, 22:09, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On May 14, 3:17 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > Which of the following statements is not true > > > > > Every element of R is different from p. > > > > > Every node of p is in some element of R > > > > Back to the facts: > > > Find a path p and a set of paths R such that > > p is not in R, but every node in p is in some element of R. > <snip attempt to change subject > > > > Hence your set R is not suitable to demonstrate your claims. > > > Does not follow. Which of these statements is not true. > > > Every element of R is different from p. > > That is correct, but uninteresting. > > > > > Every node of p is in some element of R > > That is wrong if p is different from every element of R. Does not follow (we are talking about nodes not paths and p and the elements of R are paths) The question now is Is every node of p in some element of R? For any natural number n, let z(n) be a zero node in position n. Let Z be the set of nodes, {z(n) | n a natural number} Which of the following statments is wrong? The set of nodes in p is Z Every element of Z is contained in a path in R. - William Hughes
From: Dik T. Winter on 18 May 2007 22:20
In article <1179398355.204677.123430(a)o5g2000hsb.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 17 Mai, 03:03, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > > It is not inaccessible to me. > > > > > > So you saw my answer? > > > > No. I say it is not inaccessible to me. Not that I do access it. > > > > > This set F of functions can be bijected with the > > > set R of reals. The reals form an intercession with the rationals Q. > > > The rationals form a bijection with the naturals N. > > > > Sorry, makes no sense without context. > > The context was your due posting. Either you recall it or you access > my group or yu vcannot get an answer on your question which uttered an > unjustified doubt. So you refuse to post an answer in the forum where I asked the question, giving proper references ... I have given you reasons *why* I do not access that group. That you ignore those reasons just shows arrogance. We have seen this same behaviour earlier by James Harris. Are you going to mimick him? I have *no* idea what you mean with "this set F of functions". > > > If you are interested in the discussion which we had about Cantor's > > > mistake or not mistake concerning his second diagonal proof, you may > > > enter > > >http://groups.google.com/group/sci.math.research/browse_frm/thread/5d... > > > > If I look correctly, again, it is not the case that Cantor made an error. > > It is my opinion that what I wrote about it was right, the interpretation > > being that Cantor gave an additional prove that there are sets with > > cardinality larger than the naturals. But in essence this is quite > > irrelevant. Cantor may have erred at times, that is *not* related to the > > current ways of set theory. > > It is. Set theory is simply biased. Consider the list > > 0.666... > 0.3666... > 0.33666... > 0.333666... > ... > > If the diagonal number is defined by "replace 6 by 3", then we have > two answers none of which can be preferred by logic, but the second of > which is suppressed by convention. But, again, that is *not* the diagonal proof of Cantor. And even with that notation you write nonsense. "Replace 6 by 3" yields the sequence 0.33333..., which is not in the list. I have no idea what the second answer would be. > 1) Every entry of the list differs at some place from the diagonal > number. > 2) Every initial segment of the diagonal number is represented by an > entry of the list. Both are true, if you replace (2) by: 2) Every initial segment of the diagonal number is represented by the initial segment of an entry of the list. To wit: 0.333666... does *not* represent 0.333, or you have a very strange interpretation of the word "represent". -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |