Cantor Confusion
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From: Virgil on 12 Oct 2006 14:39 In article <1160640069.503756.100380(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> > > "Albrecht" <albstorz(a)gmx.de> writes: > > > David Marcus schrieb: > > ... > > > > I don't follow. How do you know that the procedure that you gave > > > > actually "defines/constructs" a natural number d? It seems that you > > > > keep > > > > adding more and more digits to the number that you are constructing. > > > > > > What is the difference to the diagonal argument by Cantor? > > > > That a (to the right after a decimal point) infinite string of decimal > > digits defines a real number, but that a (to the left) infinite string > > of decimal digits does not define a natural number. > > And why is this so? Because an infinite string of digits is not at all > defined. Only by the factors 10^(-n) this is veiled. The due digits > become more and more unimportant because their contributions to the > number size are pulled down by the increasing exponents. But this has > been forgotten by Cantor whose diagonal proof attaches the same weight > to every digit. That is obviously wrong. Cantor merely assumes, as do most mathematicians, that in mathematics, as contrasted with physics, there need not ever be a last significant digit in a decimal expansion. HdB assumes otherwise, but has not the power to impose his assumptions on others.
From: David Marcus on 12 Oct 2006 14:42 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > Please state an internal contradiction of set theory. Please use the > > standard language of set theory/mathematics so that we can understand > > what the contradiction is without needing to ask what all the words > > mean. > > Good heavens, there are so many. Where shall I start with? > Consider the binary tree which has (no finite paths but only) infinite > paths representing the real numbers between 0 and 1. The edges (like a, > b, and c below) connect the nodes, i.e., the binary digits. The set of > edges is countable, because we can enumerate them > > 0. > /a\ > 0 1 > /b\c /\ > 0 1 0 1 > ............. > > Now we set up a relation between paths and edges. Relate edge a to all > paths which begin with 0.0. Relate edge b to all paths which begin with > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > a is inherited by all paths which begin with 0.00, the other half of > edge a is inherited by all paths which begin with 0.01. Continuing in > this manner in infinity, we see that every single infinite path is > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related to any > other path. Are you using "relation" in its mathematical sense? Please define your terms "half an edge" and "inherited". > The set of paths is uncountable, but as we have seen, it > contains less elements than the set of edges. Cantor's diagonal > argument does not apply in this case, because the tree contains all > representations of real numbers of [0, 1], some of them even twice, > like 1.000... and 0.111... . Therefore we have a contradiction: > > Card(R) >> Card(N) > || || > Card(paths) =< Card(edges) -- David Marcus
From: Virgil on 12 Oct 2006 14:45 In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: My purpose was to explain to you why your unreflected assumption is > It is not > contradictory to say that in a finite set of numbers there need not be > a largest. It seems that this false assumption is one of the basic > reasons for set theory. With any common meaning of "numbers" short of complexes, it is contradictory in mathematics, whatever it may be in "Mueckenh"'s philosophy. While it may not be possible to determine which of that finite set of numbers is largest, there has to be one.
From: Virgil on 12 Oct 2006 14:47 In article <1160648272.903391.136540(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160561733.901224.261070(a)m73g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > In article <1160397914.738238.238220(a)m7g2000cwm.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > Yes, but the assertion of Fraenkel and Levy was: "but if he lived > > > > > forever then no part of his biography would remain unwritten". That is > > > > > wrong, because the major part remains unwritten. > > > > > > > > What part? > > > > > > That part accumulated to year t, i.e., 364*t. > > > > For any give t, there will be a time t_1 at which time the events of t > > will have been written down. > > But there will never be a time at which we could say: there does not > remain any part unwritten. So? > > > > Or does "Mueckenh" posit an only finitely remote end to time? > > > > > > If you think Lim {t-->oo} 364*t = 0, we need not continue to discuss. > > > > What I think is that there is always some time t_1 enough larger than > > any t so that the events of time t are written down by time t_1. > > But what you forget is that connected with this t_1 there is an even > larger amount of unwritten days. For each t_n1, there is a t_{+1}, ad infinitum.
From: Virgil on 12 Oct 2006 14:52
In article <1160648741.707624.62340(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160577085.758246.228800(a)e3g2000cwe.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > If discontinuous functions were easily allowed everywhere, why then do > > > you think that > > > lim{n-->oo} n < 10 > > > or > > > lim{n-->oo} 1/n > 10 > > > would be wrong? > > > > Since N is not normally considered to be a topological space, continuity > > of functions with a non-topological domain N is a contradiction in > > terms. > > Apply your knowledge to the balls of the vase. Which knowledge tells me that at noon each and every ball has been removed from the vase. > > > > On the other hand, limits of real sequences (functions from N to R) have > > been quite adequately defined. One such definition is: > > Give f:N --> R and L, then > > lim_{n in N} f(n) = L (or lim_{n --> oo} f(n) = L > > is defined to mean > > For every real eps > 0, Card({n: Abs(f(n)-L) > eps}) is finite. > > For the vase problem with the number n(t) of balls in the vase after t > transactions we can find always a positive eps such that for t > t_0: > 1/n(t) < eps, hence n(t) larger than an arbitrary positive number. But that analysis does not carry beyond the times of transition, and those times do not include noon or go past noon. You are assuming properties not given. > > Therefore, your assumption of lim {t-->oo} n(t) = 0 is absurd. Your assumption that some ball that has been removed has not been removed is even more absurd. |