Ian Goddard (
Mon, 23 Feb 1998 14:53:54 -0500

The sum of all logical and space-time units equals = 0


(-) Z E R O - S U M (+)


O F M A T H & P H Y S I C S

(c) 1998 Ian Williams Goddard


By proving (a) that the nature of the identity
of the number one is binary and as such is com-
posed of two symmetrical attributes the sum of
which equals zero, (b) that all numbers derive
their primary identity by relation to zero, (c)
that all non-zero numbers are subsets of zero,
and (d) that the source and the sum of all num-
bers and of all difference between all numbers
is zero, it is proven that zero is the supreme
number, the Alpha and Omega, and the universal
constant from which there can be no deviation.
Zero-sum law expresses universal equilibrium.



The primary identity relation forming the basis
of all mathematics is the relation of 0 and 1.
The number 1 is the symbolic arising of identity
establishing the foundation of difference, rela-
tion, and identity. To understand logic and the
truth we must first understand the 0,1 relation.

If there was no difference between 0 and 1, then
there would be no number 1, thus it follows that
the identity 1 is the difference between 0 and 1
and the difference between 0 and 1 is what 1 is.
The difference between 0 and 1 is symmetrical ,
and thus 1 is 1 more (+1) than 0 and 0 is 1 less
(-1) than 1. Here's that symmetrical difference:

The Difference That Is One

0--->1 = +1
0<---1 = -1
0<-->1 = 0

1 is 1 from 0 and 0 is -1 from 1
The difference that is one
is zero in total

The double arrows express the whole 0,1 relation.
As we can see, the difference between 0 and 1 is
symmetrical, and therefore, as the identity of 1
IS this difference, the number 1 is symmetrical.
The difference that is 1 is not more +1 than -1,
it is equally both, and is therefore symmetrical.
We may also prove the 0,1 symmetry as follows:

The Identity Matrix Shows
The Difference Between 0 and 1
(found at their intersections)

0 1
0 | 0 +1 |
| | = 0
1 |-1 0 |

0 is -1 relative to 1
1 is +1 relative to 0

As the difference between 0 and 1 is symmetrical,
the 0,1 relation simultaneously defines + and -1.
The identity of 1 is this symmetrical difference.
Proving that 1 is the symmetrical relation of 1
to 0 is the fact that in the identity matrix the
intersection of 1 and 1 is 0 (and when the iden-
tity matrix is expressed as logic functions, the
relation of 1 to 1 yields a null function), this
proves that 1 derives no identity attribute from
the relation of 1 to 1. The number 1 exists ex-
clusively within the 0,1 relation, and therefore
is nothing but that entire symmetrical relation.

The number 1 is, like a coin that has two sides,
a binary entity that is composed of equal yet op-
posite components (1,-1) the sum of which equals
that from which 1 arises, which is 0. This para-
dox may be referred to as the "nonarising of 1."
As 1 cannot escape 0, the number 0 is absolute.


0 = (...-1 <-->0<--> +1...) = 0

The number 0 gives us 1 and takes it away. Where
"1" is something and "0" is nothing, we can say,
in light of the nonarising of 1, that you cannot
get something (1) from nothing (0). So identity,
like energy, is conserved -- and as we will see,
this conservation law is the universal constant.
At this webpage you'll find the zero-sum of the
symmetry of 1 defined and proven via logic func-

- <---------0---------> +


- +
If the identity of 1 is symmetrical (0<-->1) yet
our notation of 1 implies only an asymmetry that
expresses either +1 or -1 but not both, then our
notation of 1 expresses only half of the actual
and true nature of the difference that is 1. If
half of the truth works for us, that's fine, but
should we decide to express only the whole truth
and noting but the whole truth, then logic dic-
tates that we must express BOTH of the sides of
the symmetrical coin that is the identity of 1.

1 = (difference) = (+)+(-)

So the notation that expresses the whole truth
about the number 1 will express the symmetrical
1,-1 structure of the difference-derived iden-
tity of 1. Let that notation simply be "1h."

0--->1 = +1
0<---1 = -1
0<-->1 = 1h

1h = (1 + (-1)) = 0

+1 expresses 50% of the symmetrical difference
that is 1, and -1 expresses the other 50%, but
only 1h expresses 100% of the difference that
is 1. Expressing the Whole Truth about the na-
ture of identity requires that the 1h notation
replace all instances where "1" and "-1" are
denoted. Here is such a uniform conversion:

Conversion Of Half Truths To Full Truth
Via Uniform Holistic Number 1h Conversion

HALF Truth: 1 * 2 + 1 = 3
convert to 1s
HALF Truth: 1 * (1+1) + 1 = (1+1+1)

convert all 1s to 1h and sum to 100% truth

FULL Truth: 1h * (1h+1h) + 1h = (1h+1h+1h)
100% Truth: 0 * 0 + 0 = 0

Notice that because 1h = (1+(-1)) and because the
definition of 1h logically (and as such properly)
defines every instance of the numbers "1" and "-1"
to be expressions of the whole difference that is
the implicit identity structure of each and every
instance of the number 1 and -1, it follows there-
fore that by definition, the number 1h implies an
infinite series of conversions of each 1 and each
-1 to 1h within the 1h statement itself. Thus the
holistic number "1h" defines the existence of an
infinite series of self-similar entities within
itself. The holistic number 1h expresses a spon-
taneous universe. 1h defines an entity that is
simultaneously all of 0, 1, -1, and infinity:

Infinite Series Of Definition-Dictated
Replacements Of The 1 and -1 Within 1h

where ( => ) = (implies)

IF +1 => 1h, and
IF -1 => 1h, and
IF 1h = (1+(-1))
THEN (1+(-1) => ((1+(-1))+(1+(-1))) =>
unto Infinity....

ERGO: 1h <=> 2h <=> 4h <=> infinity(h) = 0

The infinity of the universe arises only because
something can come from nothing IF it = (1+(-1)),
which shall be proven by universal conservation.

- <---------0---------> +



- <-----------0-----------> +

Expressing the holistic structure of the number 1
while describing his multiverse theory (which pos-
tulates a multitude of universes emerging from no-
thing), the theoretical physicist Dr. Michio Kaku
states in the May '96 issue of Astronomy magazine:

"Creating universes out of nothing may
seen to violate cherished conservation
principles, until we realize that it
takes no energy to create a universe.
If the universe is closed like a bubble,
then the energy content of its matter
is positive, while the energy of its
gravity is negative. THE SUM IS EXACTLY
ZERO (because it requires positive en-
ergy to lift an object out of its grav-
itational well, the object's gravita-
tional energy is negative). Thus it
takes NO NET energy to create new bub-
bles, which are constantly being creat-
ed in the sea of nothing. Universes
are for free." [1] [ emphasis added ]

We can express the zero-sum of a universe, and
in so doing express the nature of 1h, as such:

Pre-Universe = 0
Universe = (1+(-1))

Pre-U = U = 0

In harmony with Kaku's multiverse theory, the
holistic number 1h simultaneously describes the
zero sum of each universe in the multiverse and
of each entity found in every single universe.
The holistic number 1h defines everything for
every universe is the infinite series of 1h;
a self-similar fractalized infinity that = 0.

- <-----------0-----------> +


None of the many varieties of numbers, such as
rational, irrational, imaginary, complex, ect,
constitute a deviation from the zero-sum law and
the conservation of identity simply because all
number varieties are founded upon the primary
unit that is the identity 1, even the fractional
numbers, for the number 1 itself can be thought
of as a 1/10 (.1) fraction of the number 10.

Imaginary numbers are based upon the square root
of -1, an operation that might be said to exceed
the one-to-one relational symmetry of 0,1. That
proposition fails, however, because the holistic
number 1h logically replace all instances of 1
and -1 due to the symmetrical nature of 1 and -1,
and the square root of 1h is 0. Yet this doesn't
mean we cannot construct a system of imaginary
numbers, for it is simply a useful invention.

If we express the system of real and imaginary
numbers using the Argand diagram which displays
the real numbers along the x axis and the imagi-
nary numbers along the y axis of a Cartesian
coordinate system, we can see the symmetrical
relations of 1 and 1i (imaginary 1) to 0:

y axis

1i ^
v (1h) 1
0<----->|-- x axis

A complex number (which is a number containing
one real and one imaginary number) contains the
symmetrical relation of each of those numbers to
0 (as we can see above), and thus the sum of all
complex numbers will equal zero. So to sum this
up to the conclusion: logical dictates that no
number-type or expression can deviate from 0


(a) all numbers are naught but multiples of the
numbers 1 or -1 or of the fractions .1 or .-1,

(b) the entire nature of the identity-superstruc-
ture of each one of the numbers 1, -1, .1 and .-1
is expressed logically as a numeric symmetry that
contains both (1 and -1) or both (.1 and .-1),

(c) that symmetrical-identity structure equals 0,

it follows that...

(d) the sum of the identity of all numbers must
equal 0. (Heretofore, the real numbers are to
be representative of all varieties of numbers.)

- <-----------0-----------> +


Every number attains its primary identity attri-
bute, which is itself (such that the primary id-
entity attribute of 4 is 4), by relation to the
number 0. Yet 0 is the only number that attains
its primary attribute by relation to itself and
thus is the only absolute number. 0 is not only
the source but the sum of all numbers and the
sum of all identity attributes of all numbers.
These facts, which shall be proven below, prove
that all numbers are subordinate subsets of 0.

The Identity Matrix Shows
The Difference Between 0 and 1
(found at their intersections)

0 1
0 | 0 1 |
| | = 0
1 |-1 0 |

By relation to 0, number 1 attains its primary
identity attribute, which is 1, yet 0 attains a
secondary identity attribute -1 by relation to 1.
0 attains its primary attribute of 0 from itself.
This means that 0 can precede the existence of 1,
and thus 0 is an absolute, not relative, identity.

0 = that which attains its identity from itself.
And the identity attribute all numbers attain by
relation to themselves (as the matrix shows) is 0.
Because 0 attains its identity from 0, and both 1
and -1 attain their identities from relation to 0,
it follows that all identities are derived from a
symmetrical relation to 0, and thus: (a) 0 is the
source of all identities and (b) zero is the end
sum of all identities = Alpha & Omega. Proof:

Alpha Omega
beginning end
source sum
add all ... -3 -2 -1 0 +1 +2 +3... = 0

As we can clearly see, zero is the source and the
sum of all numbers. Here is still more proof that
also reveals the relational structure of identity
and how and why all numbers are subsets of zero:

Identity Matrices

Identity Attributes Of

-5 -4 -3 -2 -1 0 0 1 2 3 4 5
__________________ __________________
|-5 -4 -3 -2 -1 0 | 0 0 | 0 1 2 3 4 5 |
| | | |
|-4 -3 -2 -1 0 1 |-1 1 |-1 0 1 2 3 4 |
| | | |
|-3 -2 -1 0 1 2 |-2 2 |-2 -1 0 1 2 3 |
| | | |
|-2 -1 0 1 2 3 |-3 3 |-3 -2 -1 0 1 2 |
| | | |
|-1 0 1 2 3 4 |-4 4 |-4 -3 -2 -1 0 1 |
| | | |
| 0 1 2 3 4 5 |-5 5 |-5 -4 -3 -2 -1 0 |
-------------------- --------------------

All numbers listed vertically below a number along
the top are the identity attributes of that number
as derived from its relation to the numbers listed
along the side, outside the matrix box. So number
-2 is an identity of the number 4 because number 4
is 2 less than number 6. In the same way, -2 is an
identity attribute Mr. Jones if and only if Jones
is 2 feet shorter than Mr. Smith or anything.

As the identity matrices above clearly show us, the
primary identity attribute of each number, which is
itself, is derived from its relation to 0, and thus
0 is the source of all numeric identities, which is
also expressed by the fact that 0 is the beginning:

-3 -2 -1 0 +1 +2 +3

As we can also see, all the numbers listed in the
matrices, and by extension all numbers, are iden-
tity attributes of 0, and thus 0 contains all num-
bers as attributes. We can conclude that all num-
bers are subsets of the absolute number 0 because:

(a) 0 precedes all other numbers,
(b) 0 is the only absolute number,
(c) 0 is the source of all numbers,
(d) 0 is the sum of all numbers, and
(e) 0 is the sum of all identity attributes.

It follows thus that 0 contains all numbers and
thus all numbers are subsets of 0 and 0 is thus
the superset that contains all numbers. The set
of all numbers is 0:

0 = {1,-1,2,-2,3,-3...} = 0

0 = {the set of all numbers} = 0

- <-----0-----> +

In addition to the algebraic proof that the dif-
ference between any two numbers equals 0, we can
also prove the same by the rules of set theory.

The definition of "difference between two sets"
is: the difference of sets A and B is the set of
elements which belong to A but don't belong to B.
[2] The definition of an "identity set" is: the
identity set of A is the set that contains all
the attributes of the identity of A. An example:
-2 would be a member of the identity set of Mr.
Jones if Jones is 2 feet shorter than Mr. Smith,
and also of the number 4 if 4 is 2 less than 6.

To determine if two numbers, for example 1 and 4,
are different we must identify all the members of
the identity set of 1 that are not members of the
identity set of 4. Because every number contains
every other number as an identity attribute as a
consequence of its relation to every other number
(as the identity matrix proves), the identity set
of every number is equivalent to the identity set
of every other number. For example, where "--"
expresses the set notation for "difference":

The Identity Sets Of Numbers 1 and 4

1 = {0,1,-1,2,-2,3,-3,4,-4,5,-5...}
4 = {0,1,-1,2,-2,3,-3,4,-4,5,-5...}

ERGO: 1 -- 4 = null

So logic dictates that there is no difference bet-
ween the numbers 1 and 4. This is consistent with
the zero-sum of their difference, in which 1 is 3
less (-3) than 4 and 4 is 3 more (+3) than 1. How
can this be? Obviously 4 is somehow unique from 1.

The difference between 1 and 4 is simply a product
of their sequential location, 4 comes after 1 and,
as an example, 1 attains its identity attribute of
-2 from 3 but 4 attains its identity attribute of
-2 from 6. These differences do not, however, con-
stitue any deviation from zero-sum law, since all
components of these differences are symmetrical.

- <-----------0-----------> +


The Law Of Space-Time-Motion Conservation:

Every unit of forward (+) space-time-motion corre-
sponds to a symmetrical unit of backward (-) space-
time-motion such that net space-time-motion always
equals zero.

Just as the numeric unit 1 contains both +1 and -1
a unit of space-time-motion contains both +1 and -1.

Logical & Physical Symmetry

0--->1 = +1
0<---1 = -1
0<-->1 = 0

How is this physical symmetry so? The relativity
of space-time motion. If object X moves forward 1
unit of space --> relative to a coordinate system,
it is also and simultaneously true that the coor-
dinate system moves backward <-- -1 unit of space
relative to X. The sum of the net motion equals 0.
Since we measure space by motion across it, all
our measurements (when all aspects of the motion
are summed) must equal 0. For a more comprehensive

net space-motion = (--> + <--) = 0
net space-motion = ( 1 +(-1)) = 0

The very same symmetry holds for time motion. Our
motion forward in time is forward motion --> only
relative to points in time, such as this moment,
moving backward <-- into the past. Those moments
in time that travel backwards into the past are
the hash marks on the ruler of time relative to
which forward time-motion is measured as the for-
ward time-motion that it is. Without the backward
motion of moments in time into the past, the mea-
sure of forward motion toward to the future would
not be possible. Therefore, just as the number 1
is symmetrical, time-motion is symmetrical, and
thus the measurement of net time-motion equals 0
such that the complete calculation of time is:

We have just advanced one (1) second into the
future; the nature of this advance was the
passage of one second (-1) into the past.
ERGO: net time = (1+(-1)) = 0

For a more comprehensive analysis of zero time-

As we can see, the structure of space-time-motion
and of the logical entity number 1 are identical:
each primary unit of space and of time are, just
like the primary logical unit, composed of two
symmetrical components the sum of which = 0.

Showing an identity between the structure of the
primary logical unit 1 and single units of space-
time-motion is a critical aspect of proving the
validity and universality of the zero-sum law.

Additional to the zero-sum of space-time-motion,
as we read in the quotation by physicist Michio
Kaku, the net sum of all energy in the universe
equals 0. Indeed, 0 is the universal constant.

All logical and physical structures, however com-
plex and seemingly asymmetrical, are invariably
built upon the symmetrical structures of the pri-
mary 0,1 identity relation and space-time-motion,
and therefore nothing can deviate from zero-sum.

0 = (1+(-1)) + (1+(-1)) + (1+(-1)) + (1+(-1)) = 0

[1] Michio Kaku: "What Happened Before The Big
Bang," Astronomy magazine, May 1996, p. 35-41.

[2] Seymour Lipschutz: Set Theory and Related
Topics. McGraw-Hill: New York, 1964, p. 18.

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