Can N be constructed?
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From: WM on 11 Jun 2010 11:06 On 11 Jun., 16:22, 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. > > 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. > > >> 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 > > WM wrote: > > That would be correct, unless a removal is not executed before the > > next line has been established. > > You are contradicting yourself. You said above: > | ... but remove always line number n after having constructed the > | next line number n + 1. > > Now you are saying "unless a removal is not executed" at some point. > > So are you saying that your construction rule does not actually > apply to every step? Do your rules change, or perhaps thay are applied > only randomly?- I say: If my rule applies to every step, then there is always one line left. Then it cannot be true that: "There are none left". Regards, WM
From: WM on 11 Jun 2010 11:09 On 11 Jun., 16:29, David R Tribble <da...(a)tribble.com> wrote: > 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: > >> 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. > > WM wrote: > > That is wrong. Only every line *before the last one constructed* is > > erased. > > Since your construction rule states that after every line is > constructed (and a previous line is removed) another line is > always constructed, there can be no "last one" constructed. > According to your rule, there is always a next line constructed. > > Or are you saying that your construction rules do not apply to > every line? > > > Only if "all lines" can be constructed, then all lines are > > erased and are not erased. > > How can all lines be both erased and not erased? That cannot be. Therefore the precondition must be wrong. There are not "all" lines. > Does your construction rule erase a line at each step or not? Yes. > Or perhaps your construction rule apply to some steps but not > to others? It applies to every step, if there is every step. My rule applies to every step that can be done. > > > This sheds some doubt on the assertion that all lines can be > > constructed. > > That would mean that your construction rule must fail at some point. > > Would this be because your rule is somehow flawed, or is it > because your rule simply stops working after some particular line? The reason is that infinity cannot be finished. There are at every step infinitely many lines left. Reagrds, WM
From: WM on 11 Jun 2010 11:10 On 11 Jun., 16:33, 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? > > If each step is followed by another finite number step, how would > you stop at any finite step? It is not necessary to stop somewhere in order to remain in the finite domain. Regards, WM
From: WM on 11 Jun 2010 11:12 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
From: WM on 11 Jun 2010 11:14
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 |