On May 11, 4:43 pm, Virgil <vir...@ligriv.com> wrote: > In article > <105e4062-9b00-46e8-acf2-142600826...@k8g2000pbf.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > > To see a corresponding example, integrate: S_0^1 1 dx. The > > differential vanishes as b-a -> 0 but the sum over them = 1. > > The integral from a to b certainly vanishes (does to 0) as b=a -> 0, but > the "differential" does not change all, it remains dx. > --
You see how ubiquitous the notions of summation and the differential in x are, as are a and b for the bounds of integration, or here casually. Then, as delta x goes to zero: is not the area still equal to the sum of the differential areas: only for all of them? And, is it not so only for no finite nor zero value, generally, for all of them together?
Seems hypocritical to have real analysis for no finite differential, and not have non-zero infinitesimals in the reals. Of course, that is where standard real analysis doesn't depend on non-zero infinitesimals, instead as to the synthesis of the area: under the limit.
Leibniz' notation in the integral calculus, the infinitesimal analysis, with the integral bar for summation of the less-than-any- finite and non-zero differential quantities, and only all of them, well remains quite standard.
Then, to ignore the application of these tools as to simply defining the uniformly divided unit interval as to the constant differences of successive integers seems rather closed-minded.
Of course you'd be thanked to haul out all the replete applications for all of higher mathematics, as applied, solely due transfinite cardinals, why do you refuse?
The tools of higher mathematics that are of the integral calculus remain largely unaffected by notions of transfinite cardinals. Then it is a theorem of having ZF and real analysis: additivity is but countable. Is that not a theorem of objects of ZF, not consequent its axioms and simply to remain, if not relevant, not incongruous?