Can N be constructed?
in [Logic]
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From: herbzet on 10 Jun 2010 00:15 William Hughes wrote: > WM 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. > > > > After any finite number of steps you get a line > from the list. > > Look! Over There! A Pink Elephant! Oh, PLEASE, William Hughes. Don't feed the goddam troll for another 5000 posts. -- hz
From: herbzet on 10 Jun 2010 00:26 Tim Little wrote: > William Hughes 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. If dying would keep him out of sci.logic, I'm in favor of it. -- hz
From: WM on 10 Jun 2010 06:06 On 9 Jun., 23:29, David R Tribble <da...(a)tribble.com> wrote: > 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 That would be correct, unless a removal is not executed before the next line has been established. Therefore the set of remaining lines cannot be empty. Regards, WM
From: William Hughes on 10 Jun 2010 10:10 On Jun 9, 6:29 pm, David R Tribble <da...(a)tribble.com> wrote: > 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. > 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. -William Hughes
From: William Hughes on 10 Jun 2010 10:13
On Jun 10, 7:06 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 9 Jun., 23:29, David R Tribble <da...(a)tribble.com> wrote: > > > > > 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 > > That would be correct, unless a removal is not executed before the > next line has been established. > > Therefore the set of remaining lines cannot be empty. After any finite number of steps the set of remaining lines cannot be empty. Look! Over There! A Pink Elephant! After an infinite number of steps the set of remaining lines cannot be empty. - William Hughes |