*Post by Ned Latham**Post by Tom Roberts*I have no idea what Latham thinks

First truew thing you've saud in Yonks, Roberts. Wassa madder, nose

getting a bit lo9ng?

*Post by Tom Roberts**Post by Ned Latham*As far as I know, infinity has not as yet been defined as existing

in any number set,

Then you need to STUDY. Ignorance is no excuse.

Hint: trans-finite numbers, aleph-0, aleph-1, ....

Cantor invented "transfinite numbers" specifically to *avoid* infinity,

Roberts. Try to engage your brain when you read. hmm?

As far as I know, infinity has not as yet been defined as existing

in any number set, making its numerical use in mathematics impossible.

This article defines a non-Abelian Group that defines it as a number.

I call the Group Infinitor, purely for want of a better name, and

use Ç as its symbol, partly for want of a better synbol and partly

as a reminder of its intersection with the natural number set.

¤ the real number infinita, defined as the reciprocal of infinity;

¤ the natural number 1;

¤ the natural number infinity, defined as the greatest natural number

conceivable;

Division, defined by the following enumeration of allowed

operations and their outcomes (setting a = infinita and

a / a = 1, a / 1 = a, a / y = a

1 / a = y, 1 / 1 = 1, 1 / y = a

y / a = y, y / 1 = y, y / y = 1

Perhaps Infinitor can be useful in mathematical expressions that

involve limits.

There are lots of ideas about infinities.

If you haven't studied infinities for a few years,

it's probably good reading Philip Ehrlich's recent survey

that was published by the AMS and all. It's at least

a little wider than countably infinite sets and

their uncountable powersets, with Cantorian infinity.

Most people's ideas about infinity from calculus

are about the infinitesimal of the differential.

Most people's idea of infinity are probably Archimedean,

i.e., the Archimedean idea there are numerical infinities.

The "Archimedean", in the number systems: is they don't

have infinities, i.e. they're each and all finite.

If you're interested in infinitesimals the idea besides

how the integral calculus is effectively implemented in

terms of limits, calculus is called standard, then there

are concepts of infinitesimals with pretty usual properties

as you'd expect from Newton, Leibniz, Peano, Dodgson, Stolz,

Veronese, these are ideas with nilpotent (they're greater

than zero but square is zero), or just like the differential

with fluxions and the differential.

I.e., the apparatus of the infinitesimal analysis actually

is quite standard the way of the integral calculus, and

pretty much all useful analytical notions of infinitesimals

and infinity are framed in it.

Groups have unique inverses but also it's closed under the

operation, so just adding an infinity and an infinitesimal

to a group (whose product is 1) won't leave (result in) a group.

Something like zero for example usually doesn't have unique inverses.

If you add all the infinitesimal's powers and infinity's powers

that's a group.

That would be just like the group of integers to addition,

but instead the infinitesimals are negative powers of the

first infinity and infinities are powers of the infinity.

They would also have their same order on the real number

line, wherever they might be. Also it's Abelian, commutative.

The key point about the reciprocal and that the product is unity

is the linearity of the resulting functions that are continuous in it.

Extending the space with the "infinite" values and expecting to

maintain the analytical properties of the function is a matter

for functional analysis because numerical methods fall apart.

I was reading about quantization and renormalization, about

the space of differential operators, and, there's a lot going

on in operator theory that basically works up what to read out

from the spectrometer but then gets to Heaviside and Dirac

(step and impulse) as what flattening down over the quantum numbers,

it gets to the higher moment then flattens out on the higher

moment. This is about a derivation of quantum numbers read

from Molecular Structure: the Physical Approach by Brank and Speakman.

Calculus has a special place for functions that are nowhere infinite,

as are for example nowhere infinite their derivatives, classically

continuous functions or in many places functions (of a real variable).

Functions like Heaviside step and Dirac delta (impulse) are called

not-a-real-function because for example delta is basically an infinite

spike at zero with an integral defined that is its area and equals ones.

These functions are particularly tractable as functions of real functions

in the differential operators and under operator theory.

Thinking about it, I found a spiral-space-filling curve, it has some

interesting properties like it's a singularity, and that for example

it's its own derivative and anti-derivative so is most all fitting in

all the various places in operator theory that use exponentials, for

example (differential equations). (Under operators not logarithms.)

Calculus has various outstanding requirements to implement

sum-of-histories and re-normalization without making it

re-un-de-normalization.

Vitali and Hausdorff and particularly Hausdorff has a lot

going on with the measure, of, the things, the measure theory.

As you might imagine, trying to reconstruct a wave-form from

a bunch of spectrometer readings where the values were

normalized and band-limited already, doesn't leave it easy

to reconstruct original signals, i.e. waves. Ideas from

dynamical measure theory help to describe what all the noise

should add up.

Most people's idea of infinity is the biggest counting number,

besides that there's no biggest or largest or last number

available to counting, the word "infinity" is introduced

and available as "the biggest number of all biggest-less

numbers". It's understood to be outside of arithmetic,

but still project its properties for example as linearly

that as x -> oo that 1/x -> 0.

Time must go so fast it counts to infinity when we count to one.

It starts over - but, time flies.

Time flies, but it counts one infinity at a time.

This then provides for it being a continuum (continuous).

Looking at a copy of Bartholomew Price, M.A., F.R.S., F.R.A.S.

"A Treatise on Infinitesimal Calculus", 1857, it reminds of

smooth infinitesimal analysis and the idea that points,

on a line, could be contiguous, and, continuous, with an

idea that though the complete ordered field is closed to

multiplication, and so "no smallest non-zero real", ideas

like "we may in our conception of the infinitesimal Calculus

as applied to Geometry assume the line joining two consecutive

points on a circle to be straight, and represent it by a symbol

which denotes a straight line; whereas from the geometrical

definition of a circle we know that the curvature of a curve

is continuous, and that the line joining two points of it,

however near together they are, can not be straight; and thus

our symbols though representative of such straight lines,

only approximately represent them. In this case doubtless

there may be an error; an error not in the work of the calculus;

that is true and exact; but because the geometrical quantities

are not adequately expressed by the symbols; but when by means

of integration we pass from the infinitesimal element to the

finite function, then the finite function becomes the exact

and adequate representation of the geometrical quantity,

and a compensation has taken place in the act of passing

from the infinitesimal element to the finite function. On

investigation it will, I venture to think, be found that

the exactness of the Calculus has been impugned on these

and similar grounds; and there that it has been unfairly impugned:

let it be tried on its own principles; on them I venture

to say it will stand the attack. It creates its own materials,

and is subject to its own laws; let it not be condemned

because other materials, which you try to bring within its grasp,

refuse to submit to these laws."