Date: Nov 30, 2012 11:40 PM Author: ross.finlayson@gmail.com Subject: Re: Cantor's first proof in DETAILS On Nov 30, 11:42 am, FredJeffries <fredjeffr...@gmail.com> wrote:

> On Nov 30, 8:39 am, "Ross A. Finlayson" <ross.finlay...@gmail.com>

> wrote:

>

>

>

> > > You've had this function for 13 years now and you STILL can't

> > > calculate the area of a triangle with it.

>

> > Fred Jeffries who I respect: I'd like to think that's in the context

> > of modeling Dirac's delta with triangles or radial basis functions,

> > but what's important to describe of EF as plotted is this: removing

> > all the space between the integers and plotting the elements in the

> > range it would look like f(x) = x from zero to one, half a square and

> > a triangle, but the F-Sigma Lebesgue integral of EF evaluates to one

> > not one half, now that's the surprise.

>

> > EF: CDF: of the uniform distribution of the natural integers.

>

> Sorry, I can't decipher the above two paragraphs. All I see is

> 13 years and 3 math degrees and still can't calculate the

> area of a triangle

The area of a triangle is base times height over two.

A CDF ranges from zero to one over the range of the elements in the

statistical/probabilistic distribution and is increasing. A uniform

discrete distribution would have for any m, n that CDF(m+1) - CDF(m) =

CDF(n+1) - CDF(n), constant monotone. Where EF is this CDF,

putatively, 1/d = 1/d which is true, satisfying these requirements.

The notion of a uniform probability distribution over all the naturals

is not necessarily intuitive, and I described how to build one in ZFC

besides that EF has the concomitant properties of being the CDF of a

uniform distribution of the naturals.

The reference to real functions modeling Dirac's delta a.k.a. the unit

impulse function is that this function is a spike to infinity at zero,

zero elsewhere as defined in the reals, whose integral evaluates to

one. It's standardly modeled as triangles or radial basis functions

or any other function really that have area equal to one and

diminishes to point width at zero as parameterized by an unbounded

free variable. Similarly Heaviside's step is so modeled with a

parameterized arctan() and etcetera.

Here, EF's family of functions so modeling it is simply parameterized

by d as it is unbounded.

Then, I went deeper to the foundations than that. Simply working up a

mutual definition of the real numbers as constructively at once

complete ordered field, and, partially ordered ring, with, rather

restricted transfer principle, as for example we know from Cauchy/

Weierstrass and Bishop/Cheng, then, it's possible to have the

comprehension of the function as a: primitive function, in fact

_defining_ the unit line segment. A corresponding geometry of points

and spaces to complement Euclid's of points and lines is initially

defined, with a fundamental space-filling curve defining shapes via

simple properties.

Then of course there are the set-theoretic results extra the number-

theoretic results re: cardinality, an axiomless system of natural

deduction with natural definitions of sets and ordinals following

deductively gives a theory with an empty and universal set in the

dually-self-infraconsistent dialetheic and paraconsistent.

So, yeah, in the time between noting the simple construction of EF and

today, there's quite a bit of development. Dogged determination, as

it were, for me partially satisfied in a great appreciation of the

fundamental philosphical tenets.

No, I only have a Bachelor's of Science degree (in mathematics thank

you and I know computer science). The guy who wrote a dissertation to

convince soi-disant set theorists that half the integers are even has

a Ph.D. from M.I.T. He got it for writing a dissertation in set

theory that half of the integers are even.

I wonder your familiarity with Nyquist, Shannon, Huffman et alia and

how Nyquist's sampling theorems in the discrete would apply, here to

the continuous or to sets dense in the real numbers. I don't know of

much work in that area.

And Fred Jeffries, I respect you even where you claim not to make

sense of this, thank you please for not making no sense of it.

Basically these notions are very fundamental to what is continuous and

what is discrete.

So, matter as the atom is particle and wave. What then is our simple

point?

Regards,

Ross Finlayson