Discussion:
Defining infinity in a non-Abelian Group.
(too old to reply)
Ned Latham
2019-12-30 01:14:19 UTC
Permalink
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.
j***@specsol.spam.sux.com
2019-12-30 01:25:48 UTC
Permalink
Post by Ned Latham
As far as I know, infinity has not as yet been defined as existing
in any number set, making its use in mathematics impossible.
Shows how much you know.

https://en.wikipedia.org/wiki/Infinity

Second paragraph:

"In this usage, infinity is a mathematical concept, and infinite
mathematical objects can be studied, manipulated, and used just like
any other mathematical object."

Fourth paragraph:

"The mathematical concept of infinity and the manipulation of infinite
sets are used everywhere in mathematics, even in areas such as
combinatorics that may seem to have nothing to do with them."
--
Jim Pennino
Ned Latham
2019-12-30 03:21:08 UTC
Permalink
Post by j***@specsol.spam.sux.com
Post by Ned Latham
As far as I know, infinity has not as yet been defined as existing
in any number set, making its use in mathematics impossible.
Shows how much you know.
https://en.wikipedia.org/wiki/Infinity
You're not wrong. The group remains valid, however.
j***@specsol.spam.sux.com
2019-12-30 05:06:11 UTC
Permalink
Post by Ned Latham
Post by j***@specsol.spam.sux.com
Post by Ned Latham
As far as I know, infinity has not as yet been defined as existing
in any number set, making its use in mathematics impossible.
Shows how much you know.
https://en.wikipedia.org/wiki/Infinity
You're not wrong. The group remains valid, however.
Ice cream has no bones.
--
Jim Pennino
Daniel60
2019-12-30 06:38:55 UTC
Permalink
Post by j***@specsol.spam.sux.com
Post by Ned Latham
Post by j***@specsol.spam.sux.com
Post by Ned Latham
As far as I know, infinity has not as yet been defined as existing
in any number set, making its use in mathematics impossible.
Shows how much you know.
https://en.wikipedia.org/wiki/Infinity
You're not wrong. The group remains valid, however.
Ice cream has no bones.
Really!! You must get the high quality ice cream!! ;-P
--
Daniel
news18
2019-12-30 12:19:34 UTC
Permalink
Post by j***@specsol.spam.sux.com
Post by Ned Latham
Post by j***@specsol.spam.sux.com
As far as I know, infinity has not as yet been defined as existing in
any number set, making its use in mathematics impossible.
Shows how much you know.
https://en.wikipedia.org/wiki/Infinity
You're not wrong. The group remains valid, however.
Ice cream has no bones.
But is can still bend spoons and mke hard work for a cleaver.
Basil Jet
2019-12-30 05:03:40 UTC
Permalink
Post by Ned Latham
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.
¤ 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 / y = 1, y / 1 = y
1 / y = a, 1 / a = y
a / a = 1, a / 1 = a
I presume 1/1 is 1, but for it to be a group,
y/a would have to be a, and
a/y would have to be y.
--
Basil Jet recently enjoyed listening to
The Monochrome Set - 2016 - Cosmonaut
Ned Latham
2019-12-30 05:34:09 UTC
Permalink
Post by Basil Jet
Post by Ned Latham
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.
¤ 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 / y = 1, y / 1 = y
1 / y = a, 1 / a = y
a / a = 1, a / 1 = a
I presume 1/1 is 1, but for it to be a group,
y/a would have to be a, and
a/y would have to be y.
Those operations are not defined. See the heading above:
"Division, defined by the following enumeration of allowed operations
and their outcomes...".

Nothing wrong with adding 1 / 1 = 1, I suppose, but I like the symmetry
of the enumeration as it is; and it's not like Ç needs that identity
whether within operations on its elements alone or on its elements mixed
with elements in other number sets.

The other two operations, y / a and a / y would be division by
reciprocals; effectively, squaring operations.
Basil Jet
2019-12-30 13:34:33 UTC
Permalink
Post by Ned Latham
Post by Basil Jet
Post by Ned Latham
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.
¤ 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 / y = 1, y / 1 = y
1 / y = a, 1 / a = y
a / a = 1, a / 1 = a
I presume 1/1 is 1, but for it to be a group,
y/a would have to be a, and
a/y would have to be y.
"Division, defined by the following enumeration of allowed operations
and their outcomes...".
Nothing wrong with adding 1 / 1 = 1, I suppose, but I like the symmetry
of the enumeration as it is; and it's not like Ç needs that identity
whether within operations on its elements alone or on its elements mixed
with elements in other number sets.
The other two operations, y / a and a / y would be division by
reciprocals; effectively, squaring operations.
You can define your Set any way you want, but Group has a specific
meaning. Groups are associative, this isn't. Groups have closure, this
doesn't. if 1 was the Identity, then 1 / x and x / 1 would both equal x.
If a and y were inverses, then a / y would equal y / a would equal 1.

There is only one group with three members and it is abelian.
https://groupprops.subwiki.org/wiki/Groups_of_order_3
--
Basil Jet recently enjoyed listening to
Rozi Plain - 2015 - Friend
whodat
2019-12-30 19:19:40 UTC
Permalink
Post by Ned Latham
Post by Basil Jet
Post by Ned Latham
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.
¤ 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 / y = 1, y / 1 = y
1 / y = a, 1 / a = y
a / a = 1, a / 1 = a
I presume 1/1 is 1, but for it to be a group,
y/a would have to be a, and
a/y would have to be y.
"Division, defined by the following enumeration of allowed operations
and their outcomes...".
Nothing wrong with adding 1 / 1 = 1, I suppose, but I like the symmetry
of the enumeration as it is; and it's not like Ç needs that identity
whether within operations on its elements alone or on its elements mixed
with elements in other number sets.
The other two operations, y / a and a / y would be division by
reciprocals; effectively, squaring operations.
Is there any stated purpose for this exercise
Ned Latham
2019-12-30 22:57:01 UTC
Permalink
Post by whodat
Post by Ned Latham
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.
¤ 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 / 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.
Is there any stated purpose for this exercise
Just a little contribution to mathemitics. Infinity and its inverse are
now numbers (of a sort); you can use them in any of the six ways shown
above.
Ned Latham
2019-12-31 01:06:01 UTC
Permalink
Amended by the inclusion of three more operations.

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

y / (1 / a) = y = (y / 1) / a and associativity is satisified.

As can be seen, closure too is satisfied, 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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