Cantor Confusion
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From: Sebastian Holzmann on 27 Oct 2006 06:39 Sebastian Holzmann <SHolzmann(a)gmx.de> wrote: > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: >> >> Sebastian Holzmann schrieb: >> What you regard as foolish is the explanation of the axioms which seem >> to be your gospel. These axioms and their meaning have not yet changed >> (as far as I know from modern text books and from the internet page of >> T. Jech (a leading set theorist of our days)). > > Which modern text book have you read? I cannot find any > non-biographical texts on Jech's internet page. Please do elaborate (or > rather: please don't...) I have to correct myself here: I have found an "archive" of published text. Perhaps I'll comment on them later.
From: David Marcus on 27 Oct 2006 12:04 Han de Bruijn wrote: > Virgil wrote: > > Consider that the number of balls as a function of time has infinitely > > many integer jump discontinuities which cluster around noon, so that > > there is no way that the function can be continuous at noon. > > Huh! Consider the Ocean as defined by Tony Orlow. Replace the balls in a > vase by the water molecules in an ocean - what hell is the difference!? > Then use a continuous model, as is _routinely done_ with Fluid Dynamics. > And there IS a way that the function can be continuous at noon. So, you are saying that if we change the problem, we can get your answer. I don't think anyone doubts this. Not sure why you think anyone would doubt it. > But the > problem is that you mathematicians do not understand what continuity IS. > You cannot comprehend that there can be a discrete as well as continuous > description for one and the same (physical) phenomenon. I guess you missed the point that the ball and vase problem is not a physical problem. The infinite number of balls should have been a hint. -- David Marcus
From: stephen on 27 Oct 2006 13:48 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > Virgil wrote: >> In article <1161884380.178413.126960(a)i3g2000cwc.googlegroups.com>, >> mueckenh(a)rz.fh-augsburg.de wrote: >> >>>David Marcus schrieb: >>> >>>>>All the balls have been removed before noon. >>>> >>>>OK. >>>> >>>>>But more balls are in the vase. >>>> >>>>Reason? Proof? Example? Anything? >>> >>>Consider a strictly increasing sequence with non-negative >>>terms.--------- If you can. >> >> Consider that the number of balls as a function of time has infinitely >> many integer jump discontinuities which cluster around noon, so that >> there is no way that the function can be continuous at noon. > Huh! Consider the Ocean as defined by Tony Orlow. Replace the balls in a > vase by the water molecules in an ocean - what hell is the difference!? > Then use a continuous model, as is _routinely done_ with Fluid Dynamics. > And there IS a way that the function can be continuous at noon. But the > problem is that you mathematicians do not understand what continuity IS. Irrelevant insult noted. > You cannot comprehend that there can be a discrete as well as continuous > description for one and the same (physical) phenomenon. See for example > the Fluid Tube Continuum: Can you describe a continuous version of the problem where each "unit" of water has a well defined exit time? A key part of the original problem is that the time at which each ball is removed is defined and reached. This is crucial to the problem. It is not just a matter of rates. If you added balls 1-10, then 2-20, 3-30, ... but you removed balls 2,4,6,8, ... then the vase is not empty at noon, even though the rates of insertions and removals are the same as in the original problem. So you cannot just say the rate is 10 in and 1 out and base an answer on that. Another key feature of the problem is that there are truly an infinite number of balls. In the infinite case it is true that 1) you add 10 balls and remove one ball at each step 2) each ball has a specified time of removal, and is removed at that time. In a finite version, you can only satisfy one of those conditions. Which one you pick determines the answer. Stephen
From: Lester Zick on 27 Oct 2006 14:53 On 26 Oct 2006 16:18:55 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: > >> You mean I should disregard silence on your part as a preference for >> my not justifying your own opinions on the subject of your beliefs? > >Yes. Exactly. You hold the key to all that is true. Of course. I'd have mentioned it myself but I didn't want to put too fine a point on the obvious. ~v~~
From: Lester Zick on 27 Oct 2006 15:20
On Thu, 26 Oct 2006 23:42:41 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >In article <1161856793.990116.183680(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > Within the *real* numbers the limit does exist. And a decimal number is > > > nothing more nor less than a representative of an equivalence classes. > > > > So we are agian at this point: The real numbers do exist. For the real > > numbers we have LIM 10^(-n) = 0. Therefore, i ther limit n--> oo tere > > is no application of Cantor's argument. > >And again we are back at this point. You do not comprehend what I am >writing. The reason real numbers exist is that there are sequences of >rationals that come arbitrarily close to each other. Technically the reason real numbers exist is that there are straight line and curve segments. It has nothing to do with decimal expansions. > The reason that >a decimal expansion is a representative of a real number is because the >sequence of finitely terminations of that number is a sequence of >rationals that comes arbitrarily close to other such sequences and so >falls in an equivalence class. *No* limit is involved in all of this. What is the "sequence of finitely terminations" (sic) for transcendentals? Are you suggesting the sequence itself is finite? ~v~~ |