Cantor Confusion
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From: mueckenh on 11 Oct 2006 10:48 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. Answer to b) unknown, depends on representation.
From: mueckenh on 11 Oct 2006 10:53 William Hughes schrieb: > > Let set S be > > 0.1 > 0.11 > 0.111 > ... > > Note that every member of set S has a finite number > of ones. Therefore, 0.111... is not a member of set S. > > Then it is asserted that every member of set S is covered > by some member of set S, but that no single member > of set S covers every member of set S. > > Since 0.111... is not a member of set S, it is not > asserted that it is covered by a member of set S. > > > You are correct. That is > > impossible. > > > > > But this is a red herring. What we want is > > > > > > Given two sets, a linear set A and another set B, such > > > that for every x in B there is a y in A such that y covers x. > > > Then there is a y in A such that for every x in B, y > > > covers x. > > > > > > It is easy to show that this can be false, if and only if > > > A is a linear set with no largest element. I.e. it is > > > not true. > > > > What is implied? Not my assertion is false but the assertion s false > > that there are finished infinte sets. > > No, what is implied is that an infinite set with a linear ordering > might not have a largest element. That is correct. But every element of the natural numbers is finite. Hence every element covers its predecessors. If 0.111... is covered by "the whole list", then it is covered by one element. That, however, is excuded. Regards, WM
From: mueckenh on 11 Oct 2006 10:58 Dik T. Winter schrieb: > In article <1160562822.815626.82270(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > What *might* be a sensible definition of a limit for a sequence of sets of > > > naturals is, that (given each A_n is a set of naturals), the limit > > > lim{n = 1 ... oo} A_n = A > > > > Yes, in that manner the definition runs. Cantor does not write n = > > 1...oo but puts only the n (he uses nue) under the limit. But the > > meaning is clearly this one. > > > > > exists if and only if for every p in n, there is an n0, such that either > > > (1) p in A_n for n > n0 > > > or > > > (2) p !in A_n for n > n0. > > > In the first case p is in A, in the second case p !in A. > > > > > > With that definition, indeed, > > > lim{n = 1 ... oo} A_n = N, > > > but also > > > lim{n = 1 ... oo} {n + 1, ..., 10n} = 0. > > > > > > I do not think you are meaning that definition. So what *is* your > > > definition? > > > > I do *not* believe that omega exists or is a useful notion. Therefore I > > do not give a definition, but, if necessary during the discussion, I > > use the only posdsible one as given by Cantor (see above). > > So the definition I gave for a limit of a sequence of sets you agree > with? Or not? I am seriously confused. With the definition I gave, > lim{n = 1 .. oo} {n + 1, ..., 10n} = {}. Sorry, I don't understand your definition. But considering the vase and its balls, set theory seems to allow *every* limit, just according to the suitability for defending ZFC. Regards, WM
From: Dik T. Winter on 11 Oct 2006 11:01 In article <33558$452cebe1$82a1e228$3091(a)news2.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > Dik T. Winter wrote: > > > In article <b7f51$452a1029$82a1e228$25909(a)news2.tudelft.nl> > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > > > arguments why it is admissible to allow for infinite sets. > > > > Because we can think about them. > > You can also think about the existence of angels and devils. Yes, and there are many people that allow for them. So what? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Wendy Covington on 11 Oct 2006 07:23
Re: Parents Posted: Oct 11, 2006 11:20 AM Plain Text Reply I am DESPERATLY seeking advice. I am not the typical parent. I do not like rigid traditional academics although I do believe that tradition is a good thing. (Which is why I am considering religous school even though our family is agnostic) My 12 year old has always attended a Waldorf school and we have LOVED it. There is NOTHING like it!! However, my husband is a University professor and we had to relocate for a job. We are now in South Florida, Ft. Lauderdale. We enrolled our child in a Montessori school. (No offence to Montessori folks) but we do not like this education at all. Here is a fundemental problem, espically for a child not raised in Montessori, while I recognize the value and principal of "self-directed" learning or "co-op" learning. The REALITY must also be ideal. In order for a child to succeed in the socratic environment, they must be taught HOW. In other words, it has to be modeled and then the children can break away into their own groups without a teacher sitting with them. Initially though, I feel a teacher needs to be present to essentialy teach them how to utilize this environment. I won't go into the details here, although if anyone is curious I will answer any questions you may have about the variation of waldorf vs. montessori (I think it is vast and it would better serve parents if they understood this, the two are often times considered similar because they are both "alternative") However I will say this: There are two ways to arrive at a destination (making a metaphore to a car trip) you can take the highway, and the drive will lack in scenery and expirence but you will get there or you can take the scenic back road and enjoy. Up to this point my son has had the pleasure of the scenic route and he is doing very well, not just academically (Which people over-value because while important the whole person is JUST as valuable) still, he entered this new school at a 7-8 grade level, which at their school is really a 8-9 grade level. This was from waldorf. He also loves music, himself, the earth, his friends. In other words, he is not just smart, he is happy. Now, he is doing terriable and we are doing homework all night and all weekend. I have called 100 schools and searched 100 hours on the internet for a solution. I am open to anything. I found a school in miami that talks about the "whole child" It is called the Cushman School. However, this would mean spending 5 hours of car time each day, I have a one year old also and I am not sure that is a good solution for either child, to spend so much of their life in the car. Does anyone have and suggestions? Thank you!!! |