```Date: May 11, 2013 8:18 PM
Author: ross.finlayson@gmail.com
Subject: Re: Matheology 258

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 differentialin x are, as are a and b for the bounds of integration, or herecasually.  Then, as delta x goes to zero: is not the area still equalto the sum of the differential areas:  only for all of them?  And, isit not so only for no finite nor zero value, generally, for all ofthem together?Seems hypocritical to have real analysis for no finite differential,and not have non-zero infinitesimals in the reals.  Of course, that iswhere standard real analysis doesn't depend on non-zeroinfinitesimals, instead as to the synthesis of the area:  under thelimit.Leibniz' notation in the integral calculus, the infinitesimalanalysis, 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 definingthe uniformly divided unit interval as to the constant differences ofsuccessive integers seems rather closed-minded.Of course you'd be thanked to haul out all the replete applicationsfor all of higher mathematics, as applied, solely due transfinitecardinals, why do you refuse?The tools of higher mathematics that are of the integral calculusremain largely unaffected by notions of transfinite cardinals.  Thenit is a theorem of having ZF and real analysis:  additivity is butcountable.  Is that not a theorem of objects of ZF, not consequent itsaxioms and simply to remain, if not relevant, not incongruous?Seems it's let that LUB outweighs ZFC.Regards,Ross Finlayson
```