Can N be constructed?
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From: George Greene on 9 Jun 2010 15:28 > If N can be constructed, then the elements n can be constructed in the > unary system as an infinite sequence of finite sequences of 1's (i.e. > as a list of finite sequences) If pigs can fly, then 1+1 = 2. If the conclusion is true, then obviously it does not MATTER what BULLSHIT you distracted with beforehand. Every element n of N obviously CAN be represented as a horizontal string of n 1's. Again, OBVIOUSLY, these things can be listed in increasing order of length, with the nth element n on the nth row. Whether this or isn't "constructing" anything IS NOT important! Whether it is possible "to finish" this "process" IS NOT important! YOU CAN JUST STIPULATE that this list exists and then SEE WHAT HAPPENS!
From: William Hughes on 9 Jun 2010 15:54 On Jun 9, 4:16 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 9 Jun., 21:08, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On Jun 9, 2:59 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > If this question is denied, then it is impossible to construct a > > > Cantor list and it is impossible to count beyond any finite number. > > > > If N can be constructed, then the elements n can be constructed in the > > > unary system as an infinite sequence of finite sequences of 1's (i.e. > > > as a list of finite sequences) > > > > 1 > > > 11 > > > 111 > > > ... > > > > This list contains all 1's that are contained in 111... > > > The claim is that no line contains all these 1's. This claim can be > > > disproved. > > > > Proof: Construct the above list, but remove always line number n after > > > having constructed the next line number n + 1. > > > The line you get after an infinite number of steps is not > > a line from the list. > > But the list you get after an infinite number of steps (with no > predecessor line removed) does not contain that line? Yes, the set of lines you write down is not the same as the set of lines that you have constructed. The list contains the lines that you write down and do not erase. You "get" limit line after doing an infinite number of steps, however the limit line is not a line that you write down. > > > > > > Then the list shrinks to a single line > > > Yes the line 111... > > And this line is different from the line you get when not removing its > predecessors? > Nope. The limit line is the same whether or not you remove its predecessors. - William Hughes
From: WM on 9 Jun 2010 16:23 On 9 Jun., 21:54, William Hughes <wpihug...(a)hotmail.com> wrote: > On Jun 9, 4:16 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > On 9 Jun., 21:08, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > On Jun 9, 2:59 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > If this question is denied, then it is impossible to construct a > > > > Cantor list and it is impossible to count beyond any finite number. > > > > > If N can be constructed, then the elements n can be constructed in the > > > > unary system as an infinite sequence of finite sequences of 1's (i.e. > > > > as a list of finite sequences) > > > > > 1 > > > > 11 > > > > 111 > > > > ... > > > > > This list contains all 1's that are contained in 111... > > > > The claim is that no line contains all these 1's. This claim can be > > > > disproved. > > > > > Proof: Construct the above list, but remove always line number n after > > > > having constructed the next line number n + 1. > > > > The line you get after an infinite number of steps is not > > > a line from the list. > > > But the list you get after an infinite number of steps (with no > > predecessor line removed) does not contain that line? > > Yes, the set of lines you write down is not the same > as the set of lines that you have constructed. In fact I write nothing, but only construct. For insance I construct by division 1/9 = 0.111... That is a construction. And that construction can be done as 0.1 0.11 0.111 .... up to a fixed line or without end. Then I have constructed 1/9. And there is no difference whether I show it in form of a list or in form of a single line. Not the slightest difference! > The list > contains the lines that you write down and do not erase. > You "get" limit line after doing an infinite number > of steps, however the limit line is not a line that > you write down. You get the limit as a line or as a line of the list. If it is possible to get the limit. In particular if it is possible to get the limit in form of the diagonal of the Cantor list. > > > > > > > Then the list shrinks to a single line > > > > Yes the line 111... > > > And this line is different from the line you get when not removing its > > predecessors? > > Nope. The limit line is the same whether or not you > remove its predecessors. Correct. Therefore the limit exists as an infinite sequence 0.111... in one single line witghout list as well as a line of the list (i.e. when its predecessors have not been removed) as well as its diagonal - or it does not exist at all as a sequence of 1's (but only as a finite definition like 1/9). It is nonsensical to try to argue in favour of a distinction. In particular it is nonsensical to argue that all 1's are in the single line 0.111... but are not in a single line of the list. Regards, WM
From: David R Tribble on 9 Jun 2010 17:29 WM wrote: >> If N can be constructed, then the elements n can be constructed in the >> unary system as an infinite sequence of finite sequences of 1's (i.e. >> as a list of finite sequences) >> 1 >> 11 >> 111 >> ... >> >> This list contains all 1's that are contained in 111... >> The claim is that no line contains all these 1's. This claim can be >> disproved. >> >> 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. Yes, that much is certain. > Look! Over There! A Pink Elephant! > 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. 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. This is because every line is removed at some finite step in the sequence of infinite steps. There is no point in the sequence where a line (finite or otherwise) is not removed from the list. After all the steps, an infinite number of lines have been removed from the list. There are none left.
From: Tim Little on 9 Jun 2010 21:22
On 2010-06-09, William Hughes <wpihughes(a)hotmail.com> wrote: > After any finite number of steps you get a line > from the list. > > Look! Over There! A Pink Elephant! > > After an infinite number of steps you get a line from > the list. I was wondering whether WM might have died from his degenerative brain disorder. For all his faults in mathematics and as a human being, I am glad that was not actually the case. - Tim |