Cantor Confusion
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From: Virgil on 11 Oct 2006 14:39 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. 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.
From: Virgil on 11 Oct 2006 14:50 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. 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.
From: Virgil on 11 Oct 2006 14:54 In article <1160578088.974689.303450(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > [...] > > > > > There is no largest natural! There is a finite set of > > > arbitrarily large naturals. The size of the numbers is unbounded. > > > > > > > I can only conclude you have knocked youself out. > > Try the following gedanken-experiment to become accustomed with it: > a) How many different natural numbers can you store using a maximum of > 100 bits? > b) What is the largest natural number you can store with a maximum of > 100 bits? > > Regards, WM > > Answer to a) less than 100. I make the answer to A as 2^100, if each bit must be allowed to have its own personal bit position. > Answer to b) unknown, depends on representation.
From: Virgil on 11 Oct 2006 15:11 In article <452d14fe$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Dik T. Winter wrote: > > 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. > > It defines something. An infinite string of digits. but every standard natural number is defined by a finite string of digits, given a base, so those infinite string define nothing at all. Besides themselves. What do you call that? An infinite string. > If the value up to and > including every digit is finite, how can the string represetn anything > but a finite value? If a binary string s:N --> {01} is such that s(n) = 1 for all n in N, then its "value" is sum_{n in N} 2^n, which diverges. But all the partial sums, sum_{n =1..k}2^n are all finite. So the value up to and including every digit is finite and the string itself cannot represent any finite value. So another of TO's fairy tales is debunked.
From: David Marcus on 11 Oct 2006 18:30
cbrown(a)cbrownsystems.com wrote: > David Marcus wrote: > > David R Tribble wrote: > > > Dik T. Winter wrote: > > > >> And you also start with definitions, or a model. I did *not* state that > > > >> it was difficult to define, or to make a model. But without such a > > > >> definition or model we are in limbo. I think other (consistent) definitions > > > >> or models are possible, giving a different outcome. > > > > > > > > > > Virgil wrote: > > > >> Can you suggest one? One that does not ignore the numbering on the balls > > > >> as some others have tried to do. > > > > > > > > > > David Marcus wrote: > > > > Models that ignore the number of the balls are certainly models. Whether > > > > they are reasonable translations of the original problem into > > > > Mathematics is a separate question. > > > > > > I tried modeling the problem with sets: > > > B_0 = { } > > > B_1 = B_0 U {1,2,3,...,10} \ {1} > > > ... > > > B_n = B_(n-1) U {10(n-1)+1,10(n-1)+2,...,10(n-1)+10} \ {n} > > > ... > > > > > > The resulting "balls in the vase at noon" would then seem to be B_w. > > > Which is problematic, since there is no ball labeled "w". So this is > > > probably a bad model of the problem. (I don't claim to be an expert.) > > > > I think you are still using the ball labeling. So, even if your model > > gave an answer for noon, I don't think it is what Virgil asked for. > > > > You could define B_w = {x | exists m such that for all n > m, x in B_n}. > > Another possibility is B_w = {x | for all m, there is an n > m such that > > x in B_n}. > > In both your and David Tribble's examples, I don't see any reference to > what time t step n occurs. This information is required to determine > "the number of balls at time t" for /any/ time t (not just noon). Yes, it would certainly seem to help if the model explicitly included time. -- David Marcus |