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.