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.