Can N be constructed?
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From: WM on 12 Jun 2010 09:04 On 12 Jun., 03:03, "K_h" <KHol...(a)SX729.com> wrote: > > It can be finished in the limiting case. Write down 0, then > add a decimal point after it and you get 0.1. Then write > another 1 after that to get 0.11, and another 1 after that > to get 0.111, and, after ALEPH_0 markings you will have > 0.111... or 1/9. So you see, completed infinity is > necessary in order to have a decimal representation of 1/9. > That was one of the motivations the modern founders of set > theory had for embracing Cantor's ideas: completed infinity > is necessary for not only irrational numbers but for decimal > representations of rationals like 1/9. Correct. But 1/9 can be constructed, i.e., every digit can be determined by an algorithm. Same holds for sqrt(2) and pi. But the big error was to assume that an infinite sequence could also exist without an algorithm and define a number. The number would only be defined when the last digit of the sequence was known. Regards, WM
From: WM on 12 Jun 2010 09:07 On 11 Jun., 20:05, David R Tribble <da...(a)tribble.com> wrote: > WM wrote: > >> Mathematics is physics (V. A. Arnold). > > David R Tribble wrote: > >> What is the physical focal length of a diopter 0 lens? > > WM wrote: > > What a pity that Arnold died. Now we will never get the answer. > > Nevertheless, your question is physics with no doubt. > > Yes, it is a question about physics. > So what is the physical focal length of a diopter 0 lens? There are no diopter 0 lenses. Even every gravitational lens represented by a hydrogen atom has a finite focal length. Regards, WM
From: WM on 12 Jun 2010 09:07 On 11 Jun., 20:06, David R Tribble <da...(a)tribble.com> wrote: > WM wrote: > >> My proof does not show which line is left. But it shows that finished > >> infinity is a non-mathematical notion. Of course it is always the last > >> line that is left, and it is impossible to get rid of a last line, > >> though the contents of the last line may change as often as desired. > > David R Tribble wrote: > >> Non sequitur. If the contents of the last line changes, then > >> obviously that must be a different last line. There must therefore > >> be more than one line you call the "last line". > > WM wrote: > > The last line only temporarily deserves that name. > > Then there is no last line at all. So it is. At least there is no last line - after all. Regards, WM
From: WM on 12 Jun 2010 09:09 On 11 Jun., 20:09, David R Tribble <da...(a)tribble.com> wrote: > WM wrote: > >> How would you get to an infinite number of steps when each step has > >> another finite number? > > David R Tribble wrote: > >> If each step is followed by another finite number step, how would > >> you stop at any finite step? > > WM wrote: > > It is not necessary to stop somewhere in order to remain in the finite > > domain. > > If you don't stop at any finite step, how can the list be incomplete? That is the characteristic feature of an infinite list. Regards, WM
From: WM on 12 Jun 2010 09:15
On 11 Jun., 23:05, Virgil <Vir...(a)home.esc> wrote: > WM conflates what is necessary during a process with what results after > the process is ended. > > Each step is during the process, but that does not apply to after the > whole thing is over. Infinity never ends. > > WM is, in effect, returning to something like the Zeno paradoxes, by > arguing that one cannot move from one point to another because there > must always be half the distance left cover Was that the general opinion until Cantor arrived? Regards, WM |