Can N be constructed?
in [Logic]
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From: Pol Lux on 12 Jun 2010 14:31 On Jun 12, 6:00 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > 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 You just did what, WM? Construct N in your backyard without any problem? I'm sooo impressed. Well done, WM. The world owes you.
From: Virgil on 12 Jun 2010 16:05 In article <8f38389c-d25d-4792-89b3-7cdb1936e72b(a)h13g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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. How does one ever "know" the "last digit" of a sequence which does not have a last digit, for example, 0.333...? Nevertheless, there is a well-defined number in decimal arithmetic defined by that sequence.
From: Virgil on 12 Jun 2010 16:13 In article <30ba4cb8-05ea-4bd1-a58e-53d02e9573e4(a)h13g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > 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. Then motion is impossible in WMland, as to move requires passing through infinitely many intermediate positions.
From: Virgil on 12 Jun 2010 16:18 In article <8c2ba6a5-3ec9-4c6e-acbc-a34133f97ca9(a)u7g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 11 Jun., 22:57, Virgil <Vir...(a)home.esc> wrote: > > > > > Actually that is NOT what Cantor says in his "diagonal" argument. What > > > > Cantor does say is that for every list of infinite binary sequences > > > > presented he can construct a binary sequence not in that list. > > > > > What would be the result of the construction as long as it is > > > unfinished? > > > > That is not Cantor's problem to solve. > > > > It is. Cantor stated that the process of comparing digits must be > finished instantaneously. That statement is wrong. Even if it were, it is not a part of the construction, so is irrelevant to that construction. > > > > > > > > > Look here for instance, if you want to have the opinion of an expert: > > >http://groups.google.de/group/sci.math.research/browse_frm/thread/4a8... > > > 04ad# > > > > There are 'experts' on both sides of the issue, with physicists on one > > side and mathematicians on the other. > > > > But, for example, the mathematics of coding, on which a huge amount of > > modern commerce is deeply dependant, is entirely and exclusively > > non-physics. > > > > So there are important, even essential, areas of mathematics in which > > physics has absolutely no part. > > Who does that coding? People did it originally, but now machines do it. Does WM wish to argue that since people are physical everything they do is the result of physics? Then all crimes are the fault of physics, too. Which justifies hanging physicists. And WM.
From: Virgil on 12 Jun 2010 16:21
In article <20f2652c-3219-4062-9e05-0d95bb64da02(a)i31g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 12 Jun., 18:51, David R Tribble <da...(a)tribble.com> wrote: > > WM wrote: > > >> It is not necessary to stop somewhere in order to remain in the finite > > >> domain. > > > > David R Tribble wrote: > > >> If you don't stop at any finite step, how can the list be incomplete? > > > > WM wrote: > > > That is the characteristic feature of an infinite list. > > > > Non sequitur. The list is incomplete (only) at every finite step. > > And there is no other. Maybe not to WM, but for eveyone else there are finished lists. > > > But if you don't stop at any finite step, how can the list be > > incomplete? > > That is the property of infinity. Then how is motion possible if those infinitely many steps must always remain incomplete? > > > > You say the list is incomplete, therefore there must be some > > step at which you are stopping to make it so. > > > Non sequitur. How is motion possible > > > Which step?cannot become complete. > > None. The list is incomplete after every step.? But not after all steps, which makes motion possible. |