Re: infinity

Jonathan Hoyle wrote:
> >> 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

You describe a misuse, but the results of infinitesimal analysis
exactly match those of the integral calculus.

Infinitesimals were reintroduced by a variety of purveyors prior to
Robinson in th mid-20'th century. Some include the beginning notions
of a dual representation of reals as complete ordered field and
partially ordered local ring, eg Schmieden and Laugwitz, or so I have
heard.

Countably infinite sequence of iota sum to one. It's basically in that
guise Leibniz' dx, the differential, where Robinson's are basically
Newton's fluents and fluxions, ZF is coconsistent with IST, w.r.t the
transfer principle.

Dedekind/Cauchy is inadequate to describe all reals.

The "Finlayson reals" are dually complete ordered field and contiguous
point sequence.

The "infinitessimal discussion" is not unrelated. It's key.

When you well-order the reals, then via an extension of Cantor's first
(proof of the "uncountability" of the real), nested intervals, a
well-ordered set of nested intervals is generated with a cardinality of
whatever well-ordered set you map to the reals. Via induction, that
many disjoint intervals are generated. As for each disjoint interval
of the reals a rational number distinct from the rational numbers
contained in each other, there are then that many rationals, or for
that matter any other set dense in the reals. As well, as the reals
are complete, then implications of Cantor's first extended include the
possiblities: a) the reals are not a set, and are thus inaccessible
from set theory, or b) there exist adjacent points in the normal
ordering of the reals, and Cantor's first does not hold.

I think countable vis-vis uncountable additivity is basically
equivocation about non-empty measure "zero". They're infinitesimals,
not zero, in the limit zero, but infinitesimals.

Measure theory basically says "from zero to one, the measure is one",
but cardinality is not shown to matter because the cardinality of
[0,1], [0,2] is the same, but their measure is different. It's like
saying cardinality is the reason that the asymptotic density of the
even integers is one half, when it isn't, because their cardinality is
the same and indistinct. There's the same "cardinality" of points
between zero and one and zero and ten. "Measure" is more intuitively
about "continua". If you sum values and get a positive result, some of
those values are positive. Cardinality divides itself out of measure
theory to get correct results.

What I hoped you might mention would be any "demonstrably correct
result" about measure zero sets dense in the reals.

In infinitesimal analysis, epsilon-delta, limit, fundamental theorem of
calculus, the integral bar is a big S for summation. That summation is
for some intervals basically evenly divided over the range of
integration, and there would be countably many of those, and, the
results are perfect, totally exact, meeting Euclidean geometry's
results that are easily verified via diagrams, visual presentation, and
experiment.

Ross

.

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