On Jan 5, 7:37 pm, fom <fomJ...@nyms.net> wrote: > On 1/5/2013 6:35 PM, Ross A. Finlayson wrote: > > > > > > > > > > > On Jan 4, 10:20 pm, Virgil <vir...@ligriv.com> wrote: > >> In article > >> <7850ae29-08d9-49ef-8c7b-e8979e037...@m4g2000pbd.googlegroups.com>, > >> "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > > >>> Consider the function that is the limit of functions f(n,d) = n/d, n = > >>> 0, ..., d; n, d E N. > > >> You mean the zero function? > > >> For every n, the limit of f(n,d) as d -> oo is 0, so your limit function > >> would have to be the zero function: f(n,oo) = 0 for all n. > >> -- > > > No, none of those is the zero function, and each d->oo has it so that > > d/d = 1. > > That is true. > > The problem is that as d -> oo the value at any > given fixed n -> 0. > > 2/3, 2/4, 2/5, 2/6, 2/7, 2/8, 2/9, 2/10, ... > > So, the pointwise limit of the function is zero.
lim_n->d n/d = 1
I think many or most are familiar with this intuititively, an infinite sweep from zero through to one, of values, arrayed in an equi- distributed fashion.
This is so where the line is drawn, it may be determined uniquely among lines by two points as Euclid, but it is drawn of all the points in it.
Then, here this function embodies a constant monotone progression over an infinite domain, that naturally defines a unit. From the continuum of natural integers, the unit of the continuum of real numbers, it may be seen this is as or more direct to abstract the continuous line segment directly from the integers instead of via the rational approximation. Basically there are reasonable formations of the real numbers, with suitable properties to support analysis, that aren't necessarily the dogmatic edifice.
So, the properties of the range of this function, as modeled by real functions, are that the values have a constant difference, that sums to one.
So, draw a line segment, it's from zero to one, there you have marked each real point on it: but only in passage, in the drawing of the mark, not the marking of its points: except the beginning, and end.
Countable additivity of measure is not of finite quantities, instead it is as to so it is modeled. Here, where a particular primitive element of the theory affords an alternative foundation, with suitably maintained results of the standard and as well otherwise inaccessible results, that may well match with our observations of events in nature, that is of reasonable interest and founds compelling arguments as to its placement, in foundations.
That's not to say that the pure in mathematics is only justified by its application, but, pure mathematics should describe everything in application. Here our theory of measure uses countable additivity for results. Of course many would be interested in results from the foundations with application, but, real analysis is of the countable, with measure one assigned to the unit, defined for the unit, not derived as the unit, in modern mathematics.
Then, in distinguishing real values, where there is trichotomy in the reals each is pairwise distinct. (And ordinals for a well-ordering are distinguishable by their elements.) Here in distinguishing the values of the unit interval by assigning to each a natural integer, this is done in their natural order, and uniquely so.
So: draw a line without putting pen to paper: that's at least a segment of the continuum of real numbers. Draw a line: it is the single gesture. The infinite regress of marking each point alone won't complete the line: draw it at once.
A spiral space-filling curve is the natural continuum.