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Han de Bruijn wrote:
> Randy Poe wrote:
>
> > Your infinitesimals are only used under integral signs.
>
> See if I can find a counter-example. As promised to Robert Low as well.
>
> Han de Bruijn
How about the counterexample from "Counterexamples in Real Analysis"
that there is an infinitesimal in the reals?
That's mathematician usage. Also, you might have noticed before dx and
dy being moved around quite aptly and so forth in terms of the Leibniz
notation, of those infinitesimals.
The "limit" of the, sum, is either an infinitely small value summed
over infinitely many or the wrong anwer, in the geometric
interpretation of well-known geometric identities of for example area,
or volume, for almost all functions.
Consider any function that is not a straight line in R^2, any finite
sum is only an approximation and never exact, of the area under said
curve. It is only in infinite induction that the sum of those
differential width sections is correct, not for any, only for all.
If dx isn't an infinitesimal, and S not an infinite sum, the integral
is rigorously not correct, because there are counterexamples to those
being finite, as described above.
In terms of via a restricted transfer principle infinite induction and
nonstandard reals that are contiguous on the real number line, the
reals are well-orderable, so, where the reals are integral
iota-multiples, the reals are well-orderable, else not.
There are only and everywhere reals between zero and one, inclusive.
Regards,
Ross F.
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