Cantor Confusion
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From: mueckenh on 16 Oct 2006 08:29 Virgil schrieb: > In article <1160815134.774717.182680(a)f16g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <1160675140.906009.253460(a)i42g2000cwa.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > > > > this does not imply > > > > > > > > > > there exists a single unary number M such that for every digit > > > > > position N, M covers 0.111... to position N > > > > > > > > Why shouldn't it? > > > > > > In general > > > "for all x there is a y such that f(x,y)" > > > does not imply > > > "there is a y such that for all x f(x,y)". > > > > > > To establish the latter requires proof over and above the former. > > > > I did not state that this be true in general, but it is true in a > > special case, namely for the covering of linear sets of finite > > elements. > > > What does "covering of linear sets of finite elements" mean? > The infinite set of naturals, as it exists in ZF and NBG, is a linear > set of finite elements according to the usual meanings of "linear order" > and "finite elements". Alas, it is not (actually) infinite. Regards, WM
From: mueckenh on 16 Oct 2006 08:37 William Hughes schrieb: > No. The fact that everything that is true about the infinite > must be justified in the finite, does not mean that everything > that can be justified in the finite must be true about the > infinite. > > You prove that something is true in the finite case. You > do not justify your transfer to the infinite case. Who has ever justified such a proof? In fact that is impossible because there is no infinity. Therefore all such "proofs" are false. But if we assume the existence of the infinite, then the sum of the geometric series is the most reliable entity at all. (Niels Abel: With the exception of the geometric series no series has ever been calculated precisely.) > > > The axiom of infinity > > applies to the paths. They are nothing but representations of real > > numbers. These exist according to set theory, therefore the paths > > exist too. > > > > Yes, but the question is not "do the paths exist?". There are two questions: Do the infinite paths exist and does the geometric series wit q = 1/2 have a limit? I don't need any further infinities. Regards, WM
From: mueckenh on 16 Oct 2006 08:45 Virgil schrieb: > In article <1160835084.478453.305880(a)m73g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > I do not object to the constraints of the mathematics of physics when > > > doing physics, but why should I be so constrained when not doing physics? > > > > Because whatever you are doing, you are doing something, and "doing" > > means utilizing and applying physics. > > > That may be a physicists view of the world, but by that same measure, > whenever one is doing physics, what he is really doing is math, so that > mathematicians should rule. I would say that physics (measuring reality and deriving the laws ruling reality) and mathematics are nearly the same, at least they were the same as long as mathematics was mathematical. Therefore both depend on reality. Regards, WM
From: William Hughes on 16 Oct 2006 08:46 mueck...(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > No. The fact that everything that is true about the infinite > > must be justified in the finite, does not mean that everything > > that can be justified in the finite must be true about the > > infinite. > > > > You prove that something is true in the finite case. You > > do not justify your transfer to the infinite case. > > Who has ever justified such a proof? In fact that is impossible because > there is no infinity. Therefore all such "proofs" are false. But if we > assume the existence of the infinite, then the sum of the geometric > series is the most reliable entity at all. (Niels Abel: With the > exception of the geometric series no series has ever been calculated > precisely.) If you wish to assme that infinity does not exist, knock yourself out. However, if you are trying to show that the assumption that infinity does exists leads to a contradiction you need to justify the proofs that you make using that assumption. > > > > > The axiom of infinity > > > applies to the paths. They are nothing but representations of real > > > numbers. These exist according to set theory, therefore the paths > > > exist too. > > > > > > > Yes, but the question is not "do the paths exist?". > > There are two questions: Do the infinite paths exist and does the > geometric series wit q = 1/2 have a limit? I don't need any further > infinities. > No, there is a third question: "What is the connection between the infinite paths and the limit of the series?" You have only shown a connection between finite paths and partial sums. - William Hughes
From: mueckenh on 16 Oct 2006 08:55
David Marcus schrieb: > > By the way: Every means to draw conclusions and to calculate results is > > mathematics. There is no need to prefer a certain language (unless > > there is someone who cannot speak another one). Would you assert > > Archimedes did not do mathematics, because he used only the Greek > > language and had not yet special symbols but Greek letters to denote > > numbers? > > The language of Mathematics has evolved over time. And it is going on to evolve, like mathematics itself. > > If you assert that the ball and vase problem shows that modern > Mathematics contains a contradiction, then please state the problem > using the language of modern Mathematics. "Balls" and "vases" are not > part of Mathematics, although people may use such language to talk > informally about Mathematics. I did mention already that "balls" is an abbreviation for "natural numbers" but that I prefer to use balls in order not to intermingle these numbers with the natural numbers used for the transactions t. > So, please state the ball and vase problem > using the language of Mathematics (e.g., "sets", "functions", > "integers", "reals") so that we can see what mathematical problem you > are talking about. > > What you wrote above (i.e., X(t), Y(t), Z(t)) uses the language of > Mathematics, but (as I pointed out above) is not the statement of a > problem since it doesn't end with a question. It is not the statement of a problem but the proof of a contradiction. Therefore I need no question mark. The result, written in mathematical language, is: lim {t-->oo} 9t = 0. Regards, WM |