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Topic: Simple analytical properties of n/d
Replies: 20   Last Post: Mar 11, 2013 11:01 PM

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William Elliot

Posts: 2,637
Registered: 1/8/12
Simple analytical properties of n/d
Posted: Mar 9, 2013 12:31 AM
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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.

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