Can N be constructed?
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From: Virgil on 11 Jun 2010 17:11 In article <63250984-8f6b-4615-969e-e78edc840df3(a)i28g2000yqa.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 11 Jun., 16:45, 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. > > > > 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". > > The last line only temporarily deserves that name. > > Regards, WM In moving from point A to point B, at any time during the process there are points still to be covered but at the end of the process there are no more points to be covered. But according to WM, it is impossible to move from point A to point B.
From: Virgil on 11 Jun 2010 17:15 In article <d8c16938-e6ac-4092-8e76-5c8d85ba2906(a)a30g2000yqn.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 11 Jun., 16:47, David R Tribble <da...(a)tribble.com> wrote: > > WM wrote: > > > Mathematics is physics (V. A. Arnold). > > > > What is the physical focal length of a diopter 0 lens? > > What a pity that Arnold died. Now we will never get the answer. > Nevertheless, your question is physics with no doubt. > > Regards, WM How does one encode messages so that they can be sent securely over the internet or other unsecure conduits? Math can tell you but physics can't, so that mathematics is NOT a subset of physics. And only fools like WM claim claim it to be.
From: K_h on 11 Jun 2010 21:03 "WM" <mueckenh(a)rz.fh-augsburg.de> wrote in message news:b8c1aa63-4a20-4b25-be12-257efe86135b(a)d37g2000yqm.googlegroups.com... On 11 Jun., 02:32, David R Tribble <da...(a)tribble.com> wrote: > WM wrote: > >> Proof: Construct the above list, but remove always line > >> number n after > >> having constructed the next line number n + 1. > > William Hughes wrote: > >> After any finite number of steps you get a line from > >> the list. > > David R Tribble wrote: > >> Yes, that much is certain. > > William Hughes wrote: > >> After an infinite number of steps you get a line from > >> the list. > >> The line you get after an infinite number of steps is > >> not > >> a line from the list. > > David R Tribble wrote: > >> No, I don't think so. After an infinite number of > >> steps, where > >> at each step a (finite) line is removed from the list, > >> you end > >> up with no lines at all. > > William Hughes wrote: > > Well this depends on defining what you "end up with" > > > If your definition is (the very reasonable) "you end up > > with any lines that have been written down but not > > erased", > > then you end up with no lines as every line you write > > down > > gets erased. > > > However, I think in this context saying that "you end up > > with the limit line 111..." is better. However, this > > definition has its problems. The main one is that you > > "end up with" a line that you never write down. > > > Note, however, that in neither case do you end up with > > a line from the list. > > Yes, exactly. You can't end up with a line that was erased > at some point, nor with a line that was never written at > any > point. > > It's obvious from the beginning that the line 111... does > not > exist in WM's list, so it cannot possibly be the line that > you > end up with. Saying so makes as much sense as saying that > you end up with the line 101010..., or > or saying that infinity can be finished. 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. _
From: Pol Lux on 11 Jun 2010 23:48 On Jun 9, 8:43 pm, herbzet <herb...(a)gmail.com> wrote: > WM wrote: > > > If this question is denied, > > <PLONK> > > -- > hz Of course N cannot be constructed! I tried in my backyard, 1,2,3... and the whole thing came crashing down, and I barely escaped with my life! Anyhow, not a very high fun grade on this one, because sci.math has already established a 100 times that N is not constructible (and that Cantor was stupid, and that all mathematicians are wrong, and that the foundations are riddled with errors...). Fun grade: 2/10 for repetitiveness. Sci.math should try to break new ground rather than prove and reprove all the time that N is not constructible.
From: WM on 12 Jun 2010 09:00
On 12 Jun., 05:48, Pol Lux <luxp...(a)gmail.com> wrote: > Of course N cannot be constructed! I tried in my backyard, 1,2,3... > and the whole thing came crashing down, and I barely escaped with my > life! As Fraenkel says, it might be impossible to construct R without impredicative means. But if N could not be constructed, then also the sequence (1/n) could not be constructed. But I just did it. Regards, WM |