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This slide
explores the consequences of the open/closed world problem.
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A difference is
made between open and closed sets.
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An example of an
open set is the set of all large internet sites e.g. all sites with more than
100.000 HTML pages. Characteristics of this set are:
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• it has no known limits (there exists of course
a limit at any point in time but it is not known; for a computer this is the
same as no limit).
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• its complement i.e. the set of all small
internet sites is also an open set
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• given a certain site there are three
possibilities:
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a) the site is a large site
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b) the site is a small site
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c) the size of the site is not known (will
happen frequently)
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So the ‘law of excluded middle’ does not
apply.
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• if the existence of a universal quantifier is
defined by the possibility of iteratively naming all elements of the set,
then such a quantifier does not exist for an open set. A quantifier
‘for_all_those_known’ does exist.
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In contrast a
closed set will be considered e.g. the set of all members of a club.
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• a limit exists of course for this set: it is
the number of members of the club.
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• the complement (all non-members) still is an
open set
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• the ‘law of excluded middle’ does apply: a
person is either member of the club, or he is not.
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• it is possible to name iteratively all members
of the club so a universal quantifier does exist.
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Other kinds of
sets exist e.g. the natural numbers.
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