Bill Wood wrote:
On Fri, 2006-06-23 at 09:38 -0400, Paul Hudak wrote:
But the limit of a chain IS the maximal element of the set of all
elements comprising the chain, since the LUB, in the case of a chain, is
unique, and thus we don't have to worry about choosing the "least"
element (i.e. it reduces to the upper bound, or maximal element).
There must be something additional going on, since in general the fact
that an ordered subset of a set has a LUB in the set does not imply that
the LUB is in the subset. For example, the subset {1 - 1/n : n in N+}
of Q has LUB = 1, but 1 is not an element of the subset. It would seem
that while the infinite list is the LUB of the chain of finite lists, it
is not itself a member of the chain of finite lists. So, what am I
missing?
I don't think you missed anything, just provided more detail than I
thought was needed. As the LUB is indeed an infinite object, it is not
in the set of finite, partial lists. But that's pretty common in domain
theory, and is analogous to the example that you gave.
-Paul
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