Can N be constructed?
in [Logic]
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From: David R Tribble on 11 Jun 2010 10:22 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?
From: David R Tribble on 11 Jun 2010 10:29 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? Does your construction rule erase a line at each step or not? Or perhaps your construction rule apply to some steps but not to others? > 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?
From: David R Tribble on 11 Jun 2010 10:33 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?
From: David R Tribble on 11 Jun 2010 10:45 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". Your construction rules say nothing about the contents of lines changing, only that lines are added and removed. Are you using different rules now?
From: David R Tribble on 11 Jun 2010 10:47
WM wrote: > Mathematics is physics (V. A. Arnold). What is the physical focal length of a diopter 0 lens? |