Date: Dec 3, 2012 12:27 AM
Subject: Re: Cantor's first proof in DETAILS

On Dec 2, 8:39 pm, Virgil <> wrote:
> In article
> <>,
>  "Ross A. Finlayson" <> wrote:

> > On Dec 2, 1:33 pm, Virgil <> wrote:
> > > In article
> > > <>,
> > > "Ross A. Finlayson" <> wrote:

> > > > Valid EF is just the one function, standardly modeled by those finite
> > > > initial approximations with the integers bounded above.

> > > Except that there is no "Valid EF", given Ross' definition of how it is
> > > to be obtained.

> > > > Funny, I respond to points with points and you remove them from your
> > > > replies

> > > Once garbage is identified as such, it should be put away as quickly and
> > > firmly as possible.

> > > --
> > EF is as simply described as other functions with analytical value
> > like Dirac's delta and Heaviside's step.

> The Dirac delta is not properly a function at all, according to any
> standard mathematical definition of functions, but a pseudo function
> based on properties desired of its 'integral'. wand while the Heaviside
> step function is a function, it is deliberately not a continuous one.
> But for Ross' EF to be a function, would require a non=zero value v such
> that v * Card(|N) = 1, and there is no such real number in any standard
> form of mathematics.
> Ross wants to invent a function that in standard mathematics can be
> proved not to exist.

> > So, why don't you go lamenting that Dirac and Heaviside's useful
> > results are taught daily in the curriculum of the reals.

> Dirac's is useful enough in physics, but not at all in pure mathematics.
> Heaviside's is a proper function anyway, just not continuous, but Ross'
> EF is totally useless mathematically, at least to everyone except Ross.
>   In fact I

> > encourage you to make that your personal cause, though, there's been a
> > lot of development since the 70's so there'd be some reading involved
> > to catch up to today's.

> > While you're at it, go about re-Vitali-izing measure theory, go about
> > noting Banach and Tarski's interesting results in doubling the line,
> > note that the universal set is Russellized already with the
> > compactifying sputnik of quantification, note that the real physical
> > universe as a set is all of its subsets (and Kolker's opinion on that
> > is "no opinion"), and explain why and how configuration of physics
> > experiments computes ratios of "dark" matter, ah, never mind, you've
> > claimed yourself willfully ignorant of anything extra your limited
> > standard.

> > While you're at it, bring forth any application solely due transfinite
> > cardinals, it would make you famous.  Ah, never mind, none are known
> > in the standard and you're not much for independent results (though I
> > do describe a use for transfinite cardinals in probability, and there
> > are transfinite cardinals yet simply not so fundamental to the real,
> > of course as concrete).

> > Amigos y amigas, or casually friends as I'd so address you, friends:
> > yes, Goedel does prove that there are true facts, about the objects of
> > our theory, not theorems of our standard theory:  there are true
> > theories, extra our standard theory.  Those real truths are of
> > profound interest to the philosopher.

> One of those "facts" is that any system capable of encompassing
> arithmetic is incapable of being sown to be consistent , at least within
> itself.

> > And then, yes, conscientious mathematicians interested in fundamental
> > truths of mathematics, and mathematicians in the applied as most of us
> > in our white-collar endeavors, or at least for engineers for example a
> > windowpane check or as you would, would be sincerely expressive of
> > interest in new mathematical truths they didn't have to defend from
> > the backward and willfully ignorant, and they're interested.

> > So I'll have it that yes, Virgil, even you could learn these new
> > things.

> The 'new things', like is EF, that Ross proposes are entirely outside of
> any coherent mathematics, being totally incompatible with either
> probability theory or statistics.
> Ross seems to be imagining his own private mathematical universe much
> like the  Wolkenmuekenheim of WM.
> --

No, you forget, or perhaps were unaware of, Goedel's condition being
on any "finitely axiomatized" system, vis-a-vis, say, an axiomless
system of natural deduction, or for that matter an infinitely-
axiomatized system, or here universally with a simple read-out. Then,
here with the consistency of that system, or it's inconsistent, those
are truths.

Card(N) isn't a real quantity. So what you say there is wrong, though
it's wrong twice. Though if you're interested in the real point at
infinity, well, you should be able to find description of number-
theory's point at infinity or one- or two-point compactification of
the reals and integers (in the projectively extended real numbers).
Cardinals are defined by themselves, don't be putting them where they
don't go, those aren't compatible types.

There are only and everywhere real numbers between zero and one. Here,
0 < EF(1) < 1. The arithmetic of iota-values, representing values
from the continuum, of real numbers, is different for the operations
as addition, and multiplication, simply as repeated addition (and
Presburger Arithmetic of addition of integers is complete).

Dirac's delta is regularly used in real analysis, for example in the
solutions of differential equations. Heaviside's step can be seen as
continuous, it just is horizontal from the left, vertical at the
origin, and horizontal to the right, though of course Heaviside
defines and uses it regardless of your quibble.

Then, quite coherently, no, generally.

No, these are considerations of the plain mathematical universe shared
among us, using standard definitions and working toward conciliation
of intuition and rigor, thank you.

Or: no, thank you.


Ross Finlayson