Cantor Confusion
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From: Dik T. Winter on 19 Dec 2006 21:47 In article <1166545226.008570.102300(a)n67g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > So for finte trees there are more edges than paths but for infinite > > > trees there are more paths than edges? > > > > Indeed. > > LOL. > > > > So for finte trees there are more edges than paths but for infinite > > > trees there are more paths than edges? > > > > Right. It is taking conclusions about the infinite from the finite cases > > when you think it is otherwise. > > Perhaps we cannot determine the limit of the series 1 + 1/2 + 1/4 + ... > unless we admit some sophisticated calculation of limits. But without > any such sophistication we can be sure the sum of arbitrarily many > terms of that series will be less than 28.50. And you assert that it is > greater than 100! That is what you call mathematics. But it is simply a > wrong application of human brain. The inequality holds for all finite paths. Why it also should hold for infinite paths escapes me. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 19 Dec 2006 21:50 In article <1166545419.868255.124160(a)n67g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > There are only finite numbers in the diagonal. QED. Right again. The only problem is that there are infinitely many of them. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Newberry on 19 Dec 2006 22:01 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1166495541.239288.43640(a)48g2000cwx.googlegroups.com> "Newberry" <newberry(a)ureach.com> writes: > > > Dik T. Winter wrote: > > > > In article <1166474763.304897.177520(a)80g2000cwy.googlegroups.com> "Newberry" <newberry(a)ureach.com> writes: > > > > ... > > > > > > That is only the case in finite trees. > > > > > > It fails miserably in infinite binary trees in which no path has a last > > > > > > node or last edge. > > > > > > > > > > Do you agree that the number of edges is the upper bound of the number > > > > > of paths but disagree that the edges are countable? > > > > > > > > It is the case in finite trees, but that makes it not true for infinite > > > > trees. > > > > > > So for finte trees there are more edges than paths but for infinite > > > trees there are more paths than edges? > > > > Indeed. > > LOL. > > > > So for finte trees there are more edges than paths but for infinite > > > trees there are more paths than edges? > > > > Right. It is taking conclusions about the infinite from the finite cases > > when you think it is otherwise. > > Perhaps we cannot determine the limit of the series 1 + 1/2 + 1/4 + ... > unless we admit some sophisticated calculation of limits. But without > any such sophistication we can be sure the sum of arbitrarily many > terms of that series will be less than 28.50. And you assert that it is > greater than 100! That is what you call mathematics. But it is simply a > wrong application of human brain. > > Regards, WM paths edges level 1: 2 = 2^1 2 level 2: 4 = 2^2 6 level 3: 8 = 2^3 14 level n: 2^n (not sure what the formula is) Does the ratio edges/paths converge to 2 for n --> infinity? It certainly makes it highly couterintutive that there are more paths then edges although I do not know if it generates a flat contradiction.
From: Virgil on 19 Dec 2006 22:46 In article <1166528419.265997.136400(a)n67g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > So, where is the sum 1+1+1+...+1 for infinitely many ones? > > > > If, as indicated by "1+1+1+...+1", there is a last "1" then there are > > only finitely many of them, and the sum is well-defined. > > Otherwise, such a sum is not defined or defineable. > > > > > > Where does the equation 1+1+1+...+1 = n cease to hold? > > > > When it ceases to be of form "1+1+1+...+1". > > I completely agree with you. > Because then we have no longer finitely many differences. And then we > have no longer a finite number n. Both statements are equivalent: > Finitely many finite numbers and not finitely many not finie numbers. > > > > > There can be > > > infinitely many finite right sides but not infinitely many finite left > > > sides? Miraculous mathematics. > > > > Who said that? If WM says it, it is WM asking for miracles, but > > mathematics does not ask it or claim it. > > > > Math, at least in ZFC or NBG, says there are infinitely many finite > > values for each side of "1+1+1+...+1 = n". > > Precisely that's a good reason why this insane theory should be > abolished. > Infinitely many ones added together yield an infinite number, not a > finite number as you wish to make me believe. You will believe what you want to believe, regardless do any evidence of your errors. I said that there are infinitely many equations of form "1+1+1+...+1 = n" with equal finite values on each side, but made no claim of any such equation with any infinite umber o either side. This statement is in complete compatibility with ZFC and NBG however incompatible it may be with WM's unfounded assumptions.
From: Virgil on 19 Dec 2006 23:02
In article <1166529309.558555.214420(a)t46g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1166453622.108599.302270(a)j72g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > We need not divide anything by infinite numbers, because there are only > > > finite numbers n in the sum > > > 1 + 1/2 + 1/4 + ...+ 1/2^n + ... > > > Every level of the tree has a finite number n. For finite n also 2^n is > > > finite. > > > > Infinitely many paths pass through every edge of the tree, so if any one > > of those edges is to be shared among those paths, it must be divided > > into infinitely many pieces. > > Obviously this has been done already if your statement "infinitely > many paths pass through every edge of the tree" is correct. In an infinite binary tree it is trivially true. Each edge is identifiable by a finite sequence of left and right branchings of which it is the last, but that finite sequence is the head of infinitely many infinite sequences of such branchings each of which is a path. > > > > > > Of course. There is only the finite case for natural numbers. There are > > > no infinite numbers in N. > > > > Then there should be no infinite binary tree in the first place, and all > > of WM's claims about such trees are based on his simultaneous allowing > > of infinite paths and denial of anything infinite. > > You believe that infiniteley many finite numbers exist. Why do you > doubt that the paths can share the same property? I do not, but WM seems to claim only finitely many finite naturals but allows a path made up of finite edges each only finitely far from the root node to be infinitely long. I do not choose to accept mutually contradictory statements as WM keeps doing. > > > WM wants to divide something into a countably infinite number of equal > > shares. > > > > If s is the size of one share then WM wants to have s + s + s ...= 1. > > > > What is the numerical value of s, WM? > > > 1/s is 2^n with n a natural number. What is the largest finite natural > number? If there could be such a beast, then could be no such thing as an infinite binary tree. The existence of such a tree requires the concomitant existence of infinitely many finite naturals. So WM is again swallowing camels but straining at gnats. |