An uncountable countable set
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From: mueckenh on 2 Oct 2006 10:29 Tony Orlow schrieb: > > > > Another one. Yes, that is +1. > > > > Regards, WM > > > > I dunno Wolfgang. I think you're going out on a limb to say that one is > the same as '1', and that "+" means "in addition to what's already > there". ;) Tony, I think that "1" is a short symbol for "one" and that "+1" is a short symbol for "put another one" to the present set. But I don't think that this is very important. Regards, WM
From: Han de Bruijn on 2 Oct 2006 10:30 stephen(a)nomail.com wrote: > Han.deBruijn(a)dto.tudelft.nl wrote: > >>Worse. I have fundamentally changed the mathematics. Such that it shall >>no longer claim to have the "right" answer to an ill posed question. > > Changed the mathematics? What does that mean? > > The mathematics used in the balls and vase problem > is trivial. Each ball is put into the vase at a specific > time before noon, and each ball is removed from the vase at > a specific time before noon. Pick any arbitrary ball, > and we know exactly when it was added, and exactly when it > was removed, and every ball is removed. > > Consider this rephrasing of the question: > > you have a set of n balls labelled 0...n-1. > > ball #m is added to the vase at time 1/2^(m/10) minutes > before noon. > > ball #m is removed from the vase at time 1/2^m minutes > before noon. > > how many balls are in the vase at noon? > > What does your "mathematics" say the answer to this > question is, in the "limit" as n approaches infinity? My mathematics says that it is an ill-posed question. And it doesn't give an answer to ill-posed questions. Han de Bruijn
From: mueckenh on 2 Oct 2006 10:39 Han de Bruijn schrieb: > stephen(a)nomail.com wrote: > > > Han.deBruijn(a)dto.tudelft.nl wrote: > > > >>Worse. I have fundamentally changed the mathematics. Such that it shall > >>no longer claim to have the "right" answer to an ill posed question. > > > > Changed the mathematics? What does that mean? > > > > The mathematics used in the balls and vase problem > > is trivial. Each ball is put into the vase at a specific > > time before noon, and each ball is removed from the vase at > > a specific time before noon. Pick any arbitrary ball, > > and we know exactly when it was added, and exactly when it > > was removed, and every ball is removed. > > > > Consider this rephrasing of the question: > > > > you have a set of n balls labelled 0...n-1. > > > > ball #m is added to the vase at time 1/2^(m/10) minutes > > before noon. > > > > ball #m is removed from the vase at time 1/2^m minutes > > before noon. > > > > how many balls are in the vase at noon? > > > > What does your "mathematics" say the answer to this > > question is, in the "limit" as n approaches infinity? > > My mathematics says that it is an ill-posed question. And it doesn't > give an answer to ill-posed questions. You are right, but the illness does not begin with the vase, it beginns already with the assumption that meaningful results could be obtained under the premise that infinie sets like |N did actually exist. Regards, WM
From: Ross A. Finlayson on 2 Oct 2006 11:12 Tony Orlow wrote: > Virgil wrote: > > In article <45201554(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> A set is a container, and is not one of the objects that it contains. > >> It is nothing more or less than its contents. > > > > It is determined uniquely and entirely by its contents, as stated in the > > axiom of extentionality. > > So we agree. There is nothing besides the members. Also, the members are only sets. I feel that I have been fair presenting construction of reason that enable a rational person to make their own decision that, for example, forceing uses a universal ordinal, the generic extension of N contains no elements not in N yet bijects to R, the Borel/Combinatorics impasse, and those as basically non-controversial. I see some truths in some of the posters with alternative opinions, also, I'm quite familiar with the standard viewpoint. I use my viewpoint, which I develop in part myself. These "infinities", as they are, are on the one hand various, as infinite more various than any finite set could ever be, on the other hand they each share certain properties and it seems at root they are each irregular. Then, where some "least" infinite set has a given construction, there are what are to most implicit aspects of that construction, implicit in the sense of being true and not just unstated but known, to somebody, thus true thus implicit. There is no universe in ZF where there are only sets, and not all of them. There is no set of all sets, nor set of all ordinal (nor cardinal) numbers as sets, no collection of a variety of other things that are necessary for the establishment of certain formal arguments, in ZF. If you talk about and use those things, as most do with for universal quantification over sets, then thus necessarily ZF is not sufficient and is at once contradictory. Oh, I'm not a crank. I think Goedel tells you the null axiom theory is the only possible theory, where if it's inconsistent or incomplete it's not A theory. Ross
From: Randy Poe on 2 Oct 2006 12:01
Han.deBruijn(a)DTO.TUDelft.NL wrote: > Randy Poe wrote: > > > Tony Orlow wrote: > > > Randy Poe wrote: > > > > Tony Orlow wrote: > > > >> Han de Bruijn wrote: > > > >>> Virgil wrote: > > > >>> > > > >>>> In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, > > > >>>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >>>> > > > >>>>> Randy Poe wrote, about the Balls in a Vase problem: > > > >>>>> > > > >>>>>> It definitely empties, since every ball you put in is > > > >>>>>> later taken out. > > > >>>>> And _that_ individual calls himself a physicist? > > > >>>> Does Han claim that there is any ball put in that is not taken out? > > > >>> Nonsense question. Noon doesn't exist in this problem. > > > >>> > > > >> That's the question I am trying to pin down. If noon exists, that's when > > > >> the vase supposedly empties, > > > > > > > > Why does the existence of noon imply there is a time > > > > which is the last time before noon? > > > > > > > > It doesn't. > > > > > > I never said it did. When did I say that? > > > > I was responding to Han, who said that "If noon exists, that's when > > the vase empties". > > HdB never said such a thing. You're right. So it's back to Tony. The quote is yours (Tony), so just follow the attributes back to "if noon exists, that's when the vase empties", and that's when you (Tony) said such a thing. - Randy |