Re: infinity

>> The sum of the infinite sequence converges. If you say
>> that over the naturals that the actual sum is that actual
>> value, which it is, then there are hypernaturals, and they're
>> the naturals. If instead you say it's the limit, I don't see
>> anything besides induction necessary there.

You're missing the key point here: it's only when a COUNTABLY infinite
sequence converges that you can use the limit. UNCOUNTABLY infinite
sequences of 0 sum to 0, but you get incorrect results. Probability
Theory is very close to Measure Theory in this regard. The line
segment [0,1] is the UNCOUNTABLE collection of 0-lengthed points, and
though the sum coverges to 0, the interval length is 1, so measure is
not preserved unde Uncountable Additivity. However, measure is
preserved under Countable Addivity, and this is why it is important to
understand the difference between infinite cardinalities.

<Unrelated infinitessimal discussion snipped>

>> People use the infinitesimal calculus every day for
>> "demonstrably correct results." The cardinality of the unit
>> square mile is the same as the cardinality of a rectangle a
>> billion miles by a trillion miles. Being off by an arbitrary
>> factor of 10^21 does not generally fall within the range of
>> "demonstrably correct".

Again, this is why Measure Theory allows Countable Additivity but does
not allow for Uncountable Additivity. Also, the misuse of
infinitessimals in the 18th century was replaced by the more rigorous
approaches of Bolzano and Weierstrass. Infinitessimals (along with
infinite hyper-integers) were reintroduced in a more rigorous manner by
Robinson in the 60's.

Hope that helps,

Jonathan Hoyle

.

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