Cantor Confusion
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From: Virgil on 18 Nov 2006 14:42 In article <1163856355.707119.306700(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > I have yet to see any WM "proof" that is mathematically or logically > > satisfactory. There are always hidden assumptions that are unwarranted. > > I use the "hidden assumption" that there are not more natural numbers > d_nn than natural numbers n. If you can accept that the are more > natural numbers than natural numbers, then we have different world > views which do not fit together and cannot be united. It is not hidden that every set that is bijectable with the naturals is bijectable with the naturals. > > So we should stop here. Your mathematics is not mine and vice versa. I > assume that part of my mathematics belongs to absolute truth which does > not rely on arbitrary axioms. List all of your "absolute truths" as if they were axioms and you will have a system which mathematicans can logically discuss. Keep them hidden, as you do, and you have nothing of any mathematical value whatsever. > Anyhow it is not useful that we comment the contributions of each other > any longer, because the positions are settled and fixed. > > =================== > > > If WM wisheds to claim that there is some n for which the length of the > > diagonal is NOT greater than n, let him produce it now, or forever hold > > his peace. > > I claim that there cannot be a diagonal element where lines are > lacking. That means: There are exactly as many lines as there are > diagonal elements. Which is totally irrelevant to the issue of the lengths of those finite lines in comparison to the "length" of the infinitely long diagonal. > And there are lines as long as the diagonal is. Name one. > Because without lines the diagonal cannot exist. If there were only finitely many lines, each of length equal to its line number, then there would be one and only one line of length equal to the length of the finite diagonal. When there are infinitely many finite lines, each of length equal to its line number, then the diagonal will have infinitely many columns, one for each line, and must, therefore, be longer than every finite line. Failure to recognize and acknowledge this puts WM in Limbo, mathematically at least.
From: David Marcus on 18 Nov 2006 14:45 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > If WM wisheds to claim that there is some n for which the length of the > > diagonal is NOT greater than n, let him produce it now, or forever hold > > his peace. > > I claim that there cannot be a diagonal element where lines are > lacking. That means: There are exactly as many lines as there are > diagonal elements. And, then we leap over a chasm to the next statement: > And there are lines as long as the diagonal is. > Because without lines the diagonal cannot exist. "Without lines the diagonal cannot exist" doesn't really seem to show that "there are lines as long as the diagonal is". > This "hidden assumption" leads to the necessity of aninfinite line > (number) if the diagonal (the set of numbers) is infinite. > This concluison excludes an actually infinite diagonal. Or, it shows that the hidden assumption is contradictory. -- David Marcus
From: Virgil on 18 Nov 2006 14:52 In article <1163856451.247199.105160(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Lester Zick schrieb: > > > >Sqrt(2) does exist as the diagonal of the square. But I call that an > > >idea in order to distinguish it from numbers which can be written in > > >lists and can be subject to a diagonal proof. I call only those > > >entities numbers which can be put in trichotomy with each other. > > > > I don't know what a diagonal proof and trichotomy may be. > > Excuse me. > 1) The diagonal proof by Cantor shows that any list of real numbers is > incomplete. For this purpose we use a list (= injective sequence) of > real numbers and exchange the n-th digit of the n-th number. The > changed digits put together yield a real number which differs from each > list entry at least at one position (it differs at position n from the > n-th list entry). This proof requires that all digits of numbers like > sqrt(2) do exist and can be exchanged. That assumption is wrong. WRONG AGAIN! Cantor's "diagonal" proof merely requires that the nth number in the list have, in principle even if not in practice, a determinable nth digit, which is quite a different issue. For example, it is definitely the case that for any given position n, the nth digit of sqrt(2) is, at least in principle, determinable, however impractical it might be to carry out such a determination. So WM's assumption is not, in fact, required. And WM's objections are, in fact, nonsense.
From: Virgil on 18 Nov 2006 14:55 In article <1163856520.981002.231130(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Of course all that is mathematics. > > Regards, WM Much of what WM claims is mathematics is nonsense and other parts are just not mathematics, and very little of it actually is mathematics.
From: David Marcus on 18 Nov 2006 15:01
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > I claim case iia: There is a potentially infinite sequence N = > > > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > > > treat all of its elements. In particular we can never complete this > > > set. We can never put it into a list > > > > OK. Knock yourself out. > > I read this several times from you. What does it mean? http://idioms.thefreedictionary.com/knock+yourself+out -- David Marcus |