• Defining infinity in a non-Abelian Group.

    From Ned Latham@21:1/5 to All on Sun Dec 29 19:19:43 2019
    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.

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  • From Tom Roberts@21:1/5 to Ned Latham on Wed Jan 1 21:53:21 2020
    [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

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  • From Ned Latham@21:1/5 to Tom Roberts on Wed Jan 1 22:09:36 2020
    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.

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  • From ross.finlayson@gmail.com@21:1/5 to Ned Latham on Fri Jan 3 00:02:10 2020
    On Wednesday, January 1, 2020 at 8:09:43 PM UTC-8, Ned Latham wrote:
    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.

    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."

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