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Tony Orlow wrote:
Robert, what you just said here hinges most centrally on one word: "converge".
What exactly does this word mean? In standard analysis it means that it
approaches a finite value as n->oo. But, isn't the salient point that it
approaches a *specific* *identifiable* value? In standard analysis, these are
the same thing, since only finite values can be specifically identifiable. But,
consider the notion that we can have specific indetifiable infinities and
infinitesimals in a nonstandard system. In this sense, n and 1/n "converge" to
specific infinite and infinitesimal values, respectively. If you broaden the
meaning of "converge" in this way, then isn't it possible to handle this case
in a similar manner, albeit a nonstandard one, as the finite product of an
infinite and an infinitesimal value?
You can (as we've been over) do something in NSA. But it doesn't give
you a measure on the standard naturals.
If you think it's possible to do something, then set it up. Define
your infinites and infinitesimals, put the appropriate topology
on so that convergence can be discussed, and then try to define
the sum over all integers of some infinitesimal so that you get
the answer 1.
Oh, one extra, picky condition. It has to make sense.
But I'm not holding my breath while I wait.
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