On Mar 9, 7:25 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > On Mar 9, 12:55 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> > wrote: > > > > > > > > > > > On Mar 8, 9:31 pm, William Elliot <ma...@panix.com> wrote: > > > > On Fri, 8 Mar 2013, Ross A. Finlayson wrote: > > > > Putting aside these remarkable features of n/d, as evaluated in the > > > > limit, twice, that the resulting triangle with two-sided points has > > > > exactly unit area, there is plenty to consider with regards to ran(f) > > > > for f: N -> R_[0,1]. > > > > What's R_[0,1]? So now your considering any function f:N -> [0,1] > > > and want to discuss rang f. > > > > > Obviously and directly as N is countable, were ZF consistent and R a > > > > set in ZF, vis-a-vis its having an arbitrarily large cardinal, ran(f) > > > > is countable, in as to whether ran(f) = R_[0,1]. Here, to say that > > > > ran(f) = R_[0,1], has that it would be sufficient to represent each > > > > real in R_[0,1], in particular systems of analytical interest, > > > > compared to: that it IS R_[0,1], with a concomitant following > > > > construction of R for its use as the complete ordered field (unique, > > > > up to isomorphism, in infinite ordered fields). > > > > Gibberish except for the observation that range f is countable. > > > This means that your statement range f = [0,1] is false. > > > > > Then, this is about detailing the analytical properties of n/d, with > > > > the natural integers defined, (or as above as a natural continuum of > > > > natural integers): before the properties of rings, fields, and the > > > > ordered fields and the complete ordered field are defined. Monoids in > > > > algebra are formal structures, but they are built upon a more > > > > primitive and here the primeval structure of the continuum, first as > > > > distinct integers then to the divisions and partitions of one integer, > > > > the unit, then via extensions among each, all in the positive > > > > (Bourbaki), and only then to the ring of integers, fields of fractions > > > > and rationals and reals, and at once: ring of reals. > > > > I'll skip this as you set aside n/d. > > > > > Then for ran(f) being f: N -> ran(f), and that having structural > > > > qualities for use as a building block of then higher level algebraic > > > > structures, a variety of results unavailable to the standard, are > > > > formalizable. > > > > Huh? Is that the output of Ross' random thought generator in math mode? > > > > > Is ran(f) = [0,1]? Is ran(g) for g = x for x from 0 to 1, [0,1]? Are > > > > all the ranges of functions with image [0,1] equal to [0,1]? Then, > > > > [0,1] isn't just the points in a set, it's all of those. In as to > > > > types, then, there's a notion that: ran(f) = [0,1]. > > > > Of course not, you showed that range f is countable and > > > unable to be any uncountable set. > > > Apply the antidiagonal argument to ran(f). > > .0 > > .1 > > .2 > > .3 > > .... > > the only item different from each is > > .oo > > and, ran(f) includes 1.0. > > > Apply the nested intervals argument to ran(f). > > .0 > > .1 > > ... > > The interval is [.0, .1], there's no missing element from ran(f)'s > > [0,1]. > > > The antidiagonal argument and nested intervals argument don't support > > that ran(f) =/= [0,1]. > > > In fact, remarkable among functions N -> R, is that the antidiagonal > > argument and nested intervals argument, DON'T apply to f. > > If the robot goes "can not compute", how did it compute that? > > A well-ordering of the reals exists in ZFC, but it doesn't have > uncountably many elements in the ordering, in their normal order. No > uncountable subset of the reals, is so ordered, that it is in its > normal (linear) order. But, each of its subsets, comprised of > elements, is a set, and for each element, there are uncountably many > elements more than it in the normal and well-ordering, or not, or > uncountably many less than it, of which it is greater, that they form > a set. > > Here I'll expand on the notions of forming from the natural integers > and their ratio in the limit, a rational and real unit. Now I won't > follow "gibberish" with "illiterate" or "random thought" with > "unimaginative" or "imperceptive", well I just did, but rather a more > extended and structured and in points concise detail, you deserve, if > I am to be fair in presentation and you find it lacking, for literate > and perceptive readers. > > Basically the consideration is as to the analytical character of f_ > \infty(n). We know that the founders of the infinitesimal analysis, > today known as the integral calculus or real analysis, saw it plainly > that there were infinitesimals that were the reals, or that make the > continuum. Then indeed the fluxions of Newton or nilpotent > infinitesimal differences of Leibniz are almostly exactly, from zero > to one, ran(f). So it is established the placement of the notion, of > f and ran(f), in the idea, of continuum analysis. > > Then we might look to modern algebra and a _standard_ development of > the number systems, for formalizibility, of the naturals then integers > then rationals then complete ordered field, and instead see a > development of the number systems first in the positive, of the > naturals then rationals (of the unit interval) then reals (of the unit > interval), then of the positive and negative and of the real line. > > Then with regards to an again _modern_ development of the set- > theoretic systems, with regards to the character of this number > systems as sets, it is described that constructing the number systems > in this manner, sees a different result from the foundations in as to > the concrete relations among them. > > It's a bit different than Vitali establishing there would be a > constant c such that the sum over infinity of that would be two, and > this throwing it away as "non-" measurable, instead that it is that > way. It's a bit different than Banach-Tarski showing that with these > non-measurable sets that the line can be doubled, and copied > indefinitely, instead that it doubles, to Nyquist (with regards to > fundamental theorems of information and signals) instead of ..., > Vitali, that the structures don't have their structure. That doesn't > change much in analysis, as, there aren't yet results of the non- > measurable in measure theory of import to practice. Instead as it is > founded in the countably additive, measure theory in the standard > (real analysis) could be retrofitted with a notion of f from natural > integers making the unit and then measure and fundamental theorems of > calculus follow, undisturbed in the general scheme. > > So, n/d has remarkable analytical properties, in extension of standard > real analysis and in alternation to modern set theory, and it always > will. >
Then, the analytical properties of n/d exist in theories much weaker than standard formulations of the entire number system. A notion is then that as there is less axiomatic and definitional support needed to establish relevant properties than all of real analysis, that the properties establish axiomatic and definitional support for alternative foundations of real analysis.
Here, ran(f) that that for each f_d<oo, the elements of ran(f) are dense in ran(f_d), that between any two elements of ran(f_d) there are infinitely many elements of ran(f). This is established as well simply that between any two elements of ran(f_d) there are elements of ran(f_c>d), increasing as c increases except for c = nd for which there are exactly n-1 elements of ran(f_nd) between each f_d(m) and f_d(m+1). Simply there are c%d == 0 ? c/d-1 : floor ((c+d)/d) -1 elements of ran(f_c>d) between each f_d(m) and f_d(m+1). Intuitively, these functions are linear, as defined for integers, and later as connected: linear, or step. So, in establishing via exhaustion analytical properties of the unit square, for infinitesimal analysis, that is then extended via arithmetic to a real analysis of regions in the plane. This remains in the abstract, in defining area by the unit square, then to Riemann, Lebesgue, and Stieltjes.
Then, a goal is to establish algebra generally and how to see results of the complete ordered field as reals and as Argand or the complex and then to the hypercomplex, (in space-like, time-like, and light- like definitions of basis vectors), in establishing the ordered field of rationals and complete ordered field of reals and then so on, from integer lattice points.
Then, as to how to define area, and to define an analysis of the unit square using (natural) integer points and integer ratios, and in the limit integer ratios, basically is into the definition of division. In dividing (partitioning) a function in the unit square into vertical or horizontal rectangles a la Riemann or Lebesgue, the idea is to build functions that approach real functions, then establishing the results Riemann and Lebesgue integration, all with natural integer points and ratios. Luckily it is already done this way, in the limit from delta the difference to d the differential, for the fundamental theorems of the derivative and integral calculus, all countable and as approximative in the limit and methods of exhaustion.
This isn't yet the proof that the reals as ran(f) multiplied over the line are the same fundamental structure as the complete ordered field of modern algebra, for purposes of the definition of real functions and real analysis. In the abstract, that's the direction. However, it is earlier defined in the dialog (or, sometimes, monologue, as it were) R as R^bar^dots.