Cantor Confusion
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From: David Marcus on 12 Oct 2006 02:56 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > I am sure you are able to translate brief notions like "to enter, to > > > escape" etc. by yourself into terms of increasing or decreasing values > > > of variables of sets, if this seems necessary to you. Here, without > > > being in possession of suitable symbols, it would become a bit tedious. > > > > Yes, I can translate it myself. However, that would only tell me how I > > interpret the problem. > > Hasn't it become clear by the discussion? > > I use two variables for sequences of sets. Further I use a function. I > use the natural numbers t to denote the index number. The balls are > simply the natural numbers. I speak of "balls" in order to not > intermingle these numbers with the index-numbers. > > The set of balls having entered the vase may be denoted by X(t) with. > So we have the mathematical definition: > X(1) = {1,2,3,...10}, X(2) = {11,12,13,...,20}, ... with UX = N > There is a bijection between t and X(t). t is a number and X(t) is a set. If t = 1, then your sentence says, "There is a bijection between 1 and X(1)". But, X(1) = {1,2,3,...10}. So, I don't follow. What do you mean, please? > The set of balls having left the vase is described by Y(t). So we have > the mathematical definition: > Y(1) = 1, Y(2) = {1,2}, ... with UY = N > There is a bijection between t and Y(t). > > And the cardinal number of the set of balls remaining in the vase is > Z(t). So we have the mathematical definition: > Z(t) = 9t with Z(t) > 0 for every t > 0. > There is a bijection between t and Z(t). -- David Marcus
From: David Marcus on 12 Oct 2006 02:58 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > You wrote that "A covers B" means that A has at least as many bars as B. > > > > Does "S is completely covered by at least one element of the infinite > > set of finite unary numbers" mean that S is covered by an A that has a > > finite number of bars? > > If S = 0.111... had only finite positions then it was covered by finite > numbers. All of them are in the list (these are unary representations > of the natural numbers) > 1 = 0.1 > 2 = 0.11 > 3 = 0.111 > ... > If a set of finite numbers of the list cover some number x of the list, > then always already one list number is sufficient to cover number x. If > some number y (in unary representation) is not in the list, then it > cannot be covered by any list number. Then its positions are not well > defined. This is the case with S. Sorry, but I don't know what you mean by "not well defined". I believe you said (in a previous post) that S is an infinite string of 1's. So, what do you mean that the positions of S are not well defined? -- David Marcus
From: David Marcus on 12 Oct 2006 03:01 mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > In my view we have not gotten very far. We still have > > > > > > the result that there is no list of all real numbers > > > > > > > > > > That is not astonishing, because there are only those few real numbers > > > > > which can be constructed. > > > > > > > > Few? Few compared to what. > > > > > > Compared to the assumed set of uncountably many. > > > > Funny, you claim that the term "uncountably many" has no > > meaning, but you use it. > > You believe in its meaning and in the great set R. > > > > > > > > > The real numbers that cannot > > > > be constructed? According to you they don't exist. But even > > > > these "few" real numbers cannot be listed! > > > > > > Nevertheless the diagonal proof shows only that there are elements of a > > > countable set which have not yet been constructed.\ > > > > No, it is much stronger. It shows that any list of constructable > > numbers > > is not complete. > > Because it had not been constructed. Nevertheless it shows that the > constructed number belongs to a countable set. Therefore all can be put > in bijection with N --- after the conxtruction is complete. > > > > > > > > > > > > > > > > > > > (we need to reinterpret our terms, real numbers are > > > > > > computable real numbers, and a list is a computable > > > > > > function from the natural numbers to the (computable) real > > > > > > numbers). > > > > > > > > > > > > If it gives you a warm fuzzy to say that > > > > > > "Every ball will be removed at some time before noon", > > > > > > > > > > No. To say that every ball will be removed, is wrong, because there is > > > > > not every ball. > > > > > > > > > > > > > If it gives you a warm fuzzy to say > > > > > > > > "For any natural N, the ball numbered N will be removed from > > > > the vase before noon" > > > > > > There is not "any natural" but only those which we can define. > > > > O, so there are now > > not only now but always > > > only a finite number of naturals, not even > > an arbitrarially large number. But you continue to > > prattle on about limits. > > > > > > > There is > > > a largest natural which ever will be defined. Hence mathematics in the > > > universe and in eternity has to do with only a very small sequence of > > > naturals. > > > > > > Writing 1,2,3,... is but cheating > > > > > > > If you want to deal with a system in which there is an unknown > > but largest natural, knock yourself out. > > That is nonsense. There is no largest natural! There is a finite set of > arbitrarily large naturals. The size of the numbers is unbounded. > > > But you have a long > > way to go before you are even close to being consistent. > > It is just the reality. It is impossible to have more than 10^100 > numbers represented by all the bits of the universe. > > > And don't attempt to use results from this system to say that > > results from another system are wrong. > > > > Note that according to you the ball in vase problem > > is trivial. At some time, strictly before noon we will > > reach the largest natural. After this, nothing happens. > > There is no largest natural, but certainly the vase problem is trivial, > because there are less than 10^100 balls. This discssion is only > interesting in order to find internal contradictions of set theory. 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. > That set theory is completely incapable to describe anything correctly > with regards to reality, that is obvious. -- David Marcus
From: Han de Bruijn on 12 Oct 2006 03:17 mueckenh(a)rz.fh-augsburg.de wrote: > imaginatorium(a)despammed.com schrieb: > >>The cranks universally proceed from some intuitions about the real >>world that - essentially - there are no discontinuous functions in >>physics. I don't know which "crank" you are talking about, but one of these "cranks" says that - essentially - there are no continuous functions either. Read this: http://hdebruijn.soo.dto.tudelft.nl/QED/index.htm#ft >>If there are no discontinuous functions, then it follows that >>you can "swap limits" - and in particular that if balls(t) is a >>function representing the number of balls in state IN at time t, then >>lim t->0 (balls(t)) = balls(0). If there _are_ discontinous functions, >>this clearly does not follow. This "crank" (HdB) has only said that balls(0) is undefined here. >>I do find it hilarious that one of the cranks proudly displays a graph >>of y = 1/x on his website, and seems to believe this tells us "the >>value of 1/0". Well, a little education would be no idleness in >>something or other, I don't doubt. Is y = 1/x on the website of this "crank" (HdB)? Where is it then? Ah, you mean this: http://hdebruijn.soo.dto.tudelft.nl/QED/index.htm#fe http://hdebruijn.soo.dto.tudelft.nl/QED/cylinder.htm Yes, and your strategy is to take my argument quite out of context. I have said that singularities of the kind 1/r can be renormalized in spaces with dimension D > 1. Yes, that is what I have said. And, BTW, this is a well known fact in the common renormalization (group) theory for Quantum ElectroDynamics. Google it up yourself. Han de Bruijn
From: Han de Bruijn on 12 Oct 2006 03:18
Dik T. Winter wrote: > 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? How serious is mathematics? Han de Bruijn |