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.