Defining infinity in a non-Abelian Group.

[Ned Latham]

Abstract:
As far as I know, infinity has not as yet been defined as existing
in any number set, making its 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.

It consists of three numbers and a division operator:

Numbers (in decreasing order of magnitude for clear definition):
¤ the natural number infinity, defined as the greatest natural number
conceivable;
¤ the natural number 1;
¤ the real number infinita, defined as the reciprocal of infinity;

Division, defined by the following enumeration of allowed
operations and their outcomes (setting a = infinita and
y = infinity and using infix / as the operator for clarity):

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

As can be seen, associativity is not satisfied, but closure is and
identity and inverse elements exist. Its use is restricted, but I
suggest it can be useful in mathematical expressions that include
limits at zero and infinity.

[Tom Roberts]

[Overlooking the many errors here, I have no idea what Latham thinks
this has to do with particle physics. He certainly did not mention it.]

On 12/29/19 7:19 PM, Ned Latham wrote:
> 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, ....

If you actually learn about these, you'll recognize
that I am speaking loosely.... This is a physics
newsgroup, not a math newsgroup, and I am a physicist,
not a mathematician.

> [...] a non-Abelian Group [...]

Except what you have is not a group. Of course you're not really looking
for a group, you're looking for a field. Which, of course, cannot be
non-Abelian.

Let's see how confused he gets from not knowing that
an algebraic field here is QUITE different from the
(geometric) tensor fields used in physics.

> [...] associativity is not satisfied [...]

See -- cannot be a group.

It's both amusing and sad to see someone work so hard to avoid learning
anything about the subject they are so desperately trying to write about
(and failing). Latham clearly does not know what any of the technical
words he uses actually mean. It is hard to fathom why using Google for 2
minutes is so far beyond his abilities (both here and in other
newsgroups). He is, of course, not alone in this, just one of the most
egregious.

Tom Roberts

[Ned Latham]

Tom Roberts wrote:
>
> I have no idea what Latham thinks

First truew thing you've saud in Yonks, Roberts. Wassa madder, nose
getting a bit lo9ng?

> Ned Latham wrote:
>> 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?

Abstract:

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.

It consists of three numbers and a division operator:

Numbers:

¤ 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
y = infinity and using infix / as the operator for clarity):

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.

[ross.fi...@gmail.com]

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."