Part 1


NXOR\XOR logic:

Membership is a fundamental concept of many mathematical branches.

If we define Membership logically, then a better understanding of fundamental mathematical concepts may be achieved.

1. Introduction

The Membership concept needs logical foundations in order to be defined rigorously.

Let in be "a member of ..."
Let out be "not a member of ..."

Definition 1:
A system is any framework which at least enables to research the logical connectives between in , out.

Let a thing be nothing or something.
Let x be a placeholder of a thing.

Definition 2:
x is called local if for any system A, x is in A xor x is out A returns true.

The truth table of locality is:
in out
0 0 → F
0 1 → T (in , out are not the same) = { }_
1 0 → T (in , out are not the same) = {_}
1 1 → F


Let x be nothing.

Definition 3:
x is called non-local if for any system A, x is in A nor x is out A returns true.

The truth table of non-locality when x is nothing:
in out
0 0 → T (in , out are the same) = { }
0 1 → F
1 0 → F
1 1 → F


Let x be something.

Definition 4:
x is called non-local if for any system A, x is in A and x is out A returns true.

The truth table of non-locality when x is something:
in out
0 0 → F
0 1 → F
1 0 → F
1 1 → T (in , out are the same) = {}


Let system Z be the complementation between NXOR(non-locality) and XOR(locality).

The truth table of Z is:
in out
0 0 → T (in , out are the same) = { }
0 1 → T (in , out are not the same) = { }_
1 0 → T (in , out are not the same) = {_}
1 1 → T (in , out are the same) = {}

By system Z we may fulfill Hilbert's organic paradigm of the mathematical language. Quoting Hilbert’s famous Paris 1900 lecture:

“…The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.”

Part 2:


A hidden assumption:

An interaction between different things (abstract or not) is possible, only if they share some common property known as their realm.

Without it each thing is totally disconnected from any other thing, and there is nothing beyond one.

XOR connective is the logical basis of disconnection where no more than a one thing exits simultaneously.

If something is a one of many things, then its realm is not less than a relation between XOR (the logical basis of disconnection) and NXOR (the logical basis of connection).

A set is a NXOR\XOR realm product, because the quantifier "for all…" is used in addition to the quantifier "there exist …", for example:

The standard definition of a proper subset is:

A is a proper subset of B if for all x that are members of A, x are members of B but there exist a y that is a member of B but is not a member of A.

NXOR is used as a hidden assumption of the definition above. In order to see it, let us omit the quantifier "for all…" and we get:

A is a proper subset of B if x that is a member of A, x is a member of B but there exist a y that is a member of B but is not a member of A.

Let us examine this part:

… but there exist a y that is a member of B but is not a member of A.

NXOR is used as a hidden assumption in two cases here:

Case 1: In order to distinguish x from y we need some relation between them that enables us to simultaneously compare one with the other, and this simultaneity is an NXOR connective.

Case 2: In order to distinguish A from B (and conclude that A is a proper subset B) we need some relation between them that enables us to simultaneously compare one with the other, and this simultaneity is an NXOR connective.

Simultaneity (in this case) means that A XOR B (or x XOR y) share the same realm (and it is timeless) and we get a NXOR\XOR realm.

A hidden assumption is devastating in the case of Logic and Mathematics.



NXOR:

NXOR is the logic that enables us to define the relations among abstract/non-abstract elements.

We will not find it in any book of modern Mathematics, because Modern Mathematics uses it as an hidden assumption.

NXOR is recognized as a property called memory, which enables us to connect things and research their relations.

Our natural ability to connect between objects (notated by ↔ and known as map, or function) is a hidden assumption of the current formal language.

It has to be understood that in order to define even a 1↔1 map, we need not less than a XOR product (notated as 1) and a NXOR product (notated as ↔).

Map is a connection (a NXOR product) that enables us to define the relations between more than a 1 XOR product, and we cannot go beyond 1 without ↔ between 1,1 .

The number 2 is actually the relations of our own memory as ↔ (as a function) between 1 object XOR 1 object.




Measurement:

" The earliest and most important examples are Jordan measure and Lebesgue measure,..."
(mathworld.wolfram/Measure)


Jordan measure:
"... The Jordan measure, when it exists, is the common value of the outer and inner (NXOR hidden assumption) Jordan measures of M"
(mathworld.wolfram/JordanMeasure)

So we need a common property in order to measure, and this common properly is based on NXOR connective that is related to XOR connective products, known as members.

So membership is not less than NXOR(the common) XOR(the distinct) relations, and (for example) the common value of set N is Size.


Lebesgue Measure:
" ... A unit line segment has Lebesgue measure 1; the Cantor set has Lebesgue measure 0. (mathworld.wolfram/LebesgueMeasure)

A segment is not less than A AND B (Lebesgue measure 1).

A set of disjoint elements (finite or non-finite) has a Lebesgue measure 0.

The Lebesgue measures 1 and 0 are equivalent to NXOR(non-local) and XOR(local) products of my system, but in the traditional system Lebesgue measures 1 is not an atom, but it is a XOR-only product (made of non-finite local elements).

It has to be understood that nothing can be measured beyond one without a relation between the local and the nonlocal (the concept of "many ..." does not exist without this relation.)




A proper subset: (a definition without a hidden assumption)

C is a proper subset of B only if both of them are based on property A and any C member is also a B member, but there is a B member that is not a C member.

For example: Size is a common property of both N and any proper subset of it.

Let E be any N member, which is divided by 2.

E is a proper subset of N only if the Size property is not ignored, so let us examine this mapping:


E = { 2, 4, 6, 8, 10, 12, 14, 16, 18, ... }
↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕
N = { 1, 2, 3, 4, 5, 6, 7, 8, 9, ... }

In the example above there is a 1-1 correspondence between E and N because we ignore Size as a common property of both N and E, and define the 1-1 correspondence between the notations that represent the size, by ignoring the size itself.

Here is the right mapping between E and N, where Size as a common property is not ignored:


E = { 2, 4, 6, 8, ... }
↕ ↕ ↕ ↕
N = { 1, 2, 3, 4, 5, 6, 7, 8, ... }


If (for example) notation 8 exists in E, then the size that it represents must exist in N, and only then E is a proper subset of N. By not ignoring Size as a common property of the natural numbers, we can clearly see that there is no 1-1 correspondence between E and N.

By the way, order is not important here, and the non-ordered mapping below is equivalent to the ordered map above:


E = { 8, 4, 2, 6, ... }
↕ ↕ ↕ ↕
N = { 7, 8, 3, 1, 6, 4, 2, 5, ... }


Scope:

One can ask: How, for example, positive and negative whole numbers are related to each other in such a way that cause them to immediately be present in some mapping (which prevents the existence of a 1-1 correspondence between some set and its proper subset)?

Let a scope be a set of any number system that its cardinality is the number of members that can be found around cardinal 0, according to some member.

In the case of N members, we get an asymmetric scope, because there are no negative members in N and it has a first member.

The set of integers is symmetric because it has no first member (as N has) and as a result its scope exists in both sides of cardinal 0, for example:

If {4, 2, 8, 6,-5} and {6, 2} are two sets of integers, then scope 6 is any member that can be found between 6 and -6.

The size of each member determines its existence in or out a given scope. Also order is not important.

If cardinality (which is the number of members of some set) is a common property of some pair of sets, then it is important, for example:

Let L1 be {4, 2, 8, 6,-5}

Let E1 be {6,2}

If cardinality (which is the number of members of some set) is a common property of L1 and E1, then if 6 of E1 is mapped with some member of L1, then any member that is in the scope of 6 (which is a member of L1) must immediately be present in the set of notations that represent L1, for example:



Map1 example:
L1 = { 4, 2, 6, -5, ... } (members of L1 that must immediately be present)
↕
E1 = { 6, ... }




Map2 example:
L1 = { 2, ... } (members of L1 that must immediately be present)
↕
E1 = { 2, ... }




Map3 example:
L1 = { 4, 2, 6, -5, ... } (members of L1 that must immediately be present)
↕ ↕
E1 = { 6, 2 }


One can say: "Symmetric scope (if w then –w, or if –w then w) is artificial".

My answer is: Traditional Mathematics uses exactly this symmetry in order to show that there is a 1-1 correspondence between N and W, for example:


W = { 1, -1, 2, -2, 3, -3, 4, -4, 5, ... }
↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕
N = { 1, 2, 3, 4, 5, 6, 7, 8, 9, ... }


But since W is symmetric, then the correct 1-1 map between W and N is ( for example):


W = { 0, 1, -1, 2, -2, 3, -3, ... }
↕ ↕ ↕
N = { 1, 2, 3, ... }

which saves the non-finite approaching to proportion of 2/1 between them, where both W and N are non-finite sets as well (no one of their members is their final member).

If the concept of Size is not a common property of the concept of Set, then a 1-1 mapping between some non-finite set and a non-finite part of it is not measured beyond cardinal 1.

As a result we get a 1-1 correspondence between the notations that represent the size, by ignoring the size itself.

Part 3:

Conclusion:

Traditional Mathematics works on the concept of Set only if the content of some set is translated to distinct sizes, which are measured by cardinality.

For example:

Let S be {joke, idea, tree, voice, @}

There is no common property between S members, but the size of S, which is 5.

We must not mix between 5, which is a single notation that represents number 5, where number 5 itself is exactly 5 distinct members that their order is not important.


In that case the concept of Size is a common property of the concept of Set, and each notation that is included in some set is no more than a representation of some size (in the case of {joke, idea, tree, voice, @}, each member is generalized to some size that can be found within some scope, and only then a 1-1 mapping can be extended beyond cardinal 1).

If we compare between collections of distinct sizes, than a 1-1 mapping is defined by the number of the distinct sizes that can be found between cardinal 0 and some given distinct size, which determines the number of the distinct sizes that are mapped to each other.

Galilio and Dedeking made a simple mistake when they defined a 1-1 correspondence between notations by ignore the common property that they represent (in the case of numbers, the common property is Size).

Cantor used this mistake in order to define the non-finite property of N by claiming that there is a 1-1 correspondence between N and a proper subset of it.

But as we show here, he was wrong in this case.

A non-finite set is simply a set that does not have a final member, and both E and N are non-finite sets as well, where Size_of_E/Size_of_N has a permanent ratio of 1/2.




Aleph0 from NXOR\XOR point of view:

Cantor's theorem about the Size of the non-finite is based on the notion that 1+aleph0=aleph0, or in other words Cardinality (or Size) is not changed under addition when we deal with infinitely many objects.

By Cantor, a Size that is not 0 (he called it aleph0) does not change the result under addition.

I understand the Set concept also from a NXOR point of view.

From this additional point of view no XOR product (anything that it is "a one of many …", and the Set concept is based on it) can be an NXOR product, and as a result (which is based on logic, and not on intuition) the Size of any non-finite set is logically incomplete (it cannot be an NXOR product, no matter how infinitely many elements it has).

Since the Size of a non-finite set is incomplete we cannot use Dedekind's 1-1 method in order to define the exact sizes of two non-finite sets (each one of them is an incomplete mathematical object).

Instead, we define the proportion that exists between non-finite (and logically incomplete(XOR))products (which are called sets), and the permanent proportion of aleph0+1/aleph0 is a non-local number greater than 1 (Remark: NXOR is used as a hidden assumption of the standaed concept of Set, and as a result local-only members exist beyond one).

If this proportion is important for us, we can use some notation in order to represent aleph0+1/aleph0 (for example: "Let @ be the representation of the permanent proportion aleph0+1/aleph0") but we must not mix between the notation "@" and the value that it represents, and Dedekind's 1-1 method does not distinguish between a value and its representation, and defines the map between the representations instead of between the values themselves, as I show here.



A 1-1 correspondence: (a new point of view)

A 1-1 correspondence exists between two non-finite sets (as I show here) if each set is a collection of unique objects, where each object has nothing in common with the rest of the unique objects.

In that case the 1-1 mapping is between infinitely many separated objects.

In this case each mapping is disjoint from any other mapping, and we get the ratio of 1/1 which is equivalent to a 1-1 correspondence.

But then infinitely many disjoint 1-1 mappings cannot be considered as a mapping between a set and its proper subset (because each mapping is a separated case) and all we have is infinitely many separated cases, with a 1/1 ratio.

Part 4:


Firstly, never ever apply information theory to complex natural phenomena because the results never bear any correlation to observable reality and I am sure you know that by now.
If X has simultanuasly more than one alternative, than X is in a superposition. For example:

Before X explores quantum wavicle y in order to define its exact location, X is in a superposition of experiment A (that defines y's exact energy) and exmeriment B (that defines y's exact location).

Complex systems are complex because thay have more than one alternative at a time. So the principle of Superposition is not limited by scale.



Also, please define the set N of which you speak.
I use Organic Natural Numbers, as bulding-blocks that define the Symmetrical states within any given finite cardinal:

The consistent products of the traditional 2-valued logic are based on dichotomy (XOR connective) between opposite concepts, where inconsistency exist if some conclusion is both A and its negation.

From this excluded middle point of view , XOR and NXOR cannot be paired.

If we use an included-middle 2-valued logic, then the middle is the product of a NXOR connective between opposite concepts.

As a result, the mathematical universe between XOR and NXOR is the result of opposites that complement each other instead of contradict each other (as they do in excluded-middle 2-valued logic).

Furthermore, the excluded-middle 2-valued logic is a particular and extreme case of NXOR\XOR logic, where the products of the middle are ignored, and as a result both opposite and its negation cannot be but a contradiction.

Let 0 be the opposite of 1, and vice versa.

A NOR B is true if both A and B are 0.
A AND B is true if both A and B are 1.
A NXOR B is true if A has the same value of B.
A XOR B is true if A does not have the same value of B.

NXOR\XOR logic is the complementation between A,B and from this new point of view the concept of Membership is extended beyond XOR, and our consistent products are consistent logical "off springs" of both NXOR\XOR connectives.

If NXOR is dominant than the products are symmetrical.

If XOR is dominant than the products are non-symmetrical.

For example, in NXOR\XOR logic each finite cardinal has more than a one state of symmetry among members, and as a result a multiset (a superposition among non-distinct members) and regular set (each member is distinct) are nothing but two symmetrical states of the same finite cardinal.

By using Symmetry as a first-order property of the Natural numbers, we get a "X-ray" picture of each partition, which goes beyond the cardinal/ordinal and add distinction as a first-order property of the Natural numbers:
http://www.geocities.com/complementarytheory/parti1.jpg
http://www.geocities.com/complementarytheory/parti2.jpg

...

Some examples:

The minimal atomic representation of a local element is a point {.}, where an atomic element is known as a urelement ( http://en.wikipedia.org/wiki/Urelement ) (which is a member of a set that has no sub-elements).

The minimal atomic representation of a non-local element is a segment {}.

When combined into a one object, a non-local urelement is not defined by a collection of local urelements (they are mutually independent exactly like two axioms) and a consistent mathematical universe can be defined among them.

Let us research the Natural number concept as a product of this consistent mathematical universe.

By Peano axioms there exits the natural number 1, where any number which is not 1 is greater then the previous number by exactly 1 unit.

As a result, each natural number is both a cardinal (satisfies the question “How many?”) and an ordinal (satisfies the question “In what order?”).

By combined between a collection of local urelements and a non-local urelement, we discover that the natural numbers, based on Peano axioms, are a special case of this combination.

In other words, within any given cardinal > 1 there are several states which their exact order cannot be defined because the exact identity of each element is not known yet.

It means that there is a superposition of identities that have to be broken in order to get the broken state which defined by Peano axioms (where each element is both a cardinal and an ordinal).

Here is an illustration that (I hope) can help to understand this new idea (in this case non-locality is represented by an outer arc and locality is represented by a collection of internal arcs):

http://www.geocities.com/complementarytheory/ONN1-5.jpg
Peano products are marked by the orange rectangles. The other elements are not defined by Peano axioms, because no one of them is both a cardinal and an ordinal.

By combining between locality and non-locality we get an organic mathematical structure, which is not lass than complementary relations between the whole and the parts.

Here is the standard definition of the natural numbers:

The set of all natural numbers is the set N = {x | x є I for every inductive set I}.

Thus, a set x is a natural number iff it belongs to every inductive set. Each member of an inductive set is both a cardinal and an ordinal, because it is based on a broken-symmetry bridging (bridging is an association between mutually independent states like local element(s) and a non-local urelement), represented by each blue pattern in the figure below. Armed with symmetry as a first-order property, we define a bridging that cannot be both a cardinal and an ordinal, represented by each magenta pattern in the figure below. The products of the bridging between the local and the non-local are called organic natural numbers (please open the link below):

http://www.geocities.com/complementarytheory/icmfig4.jpg

Each pattern in this figure is both local/global state of the organic natural numbers system (which is based on the associations between non-locality[represented by horizontal lines] and locality[represented by vertical lines]) as can be seen in the case of the blue broken-symmetry patterns.

When combined into a one system, locality and non-locality can be measured by parallel symmetry (where no element has a unique identity) and serial broken symmetry (where each element has a unique identity).

The transformation from parallel symmetry to serial symmetry can be used in order to research the evolution of Complexity as a product of self reference tendency.

Here is some illustration that demonstrates this idea by using the organic natural numbers: http://www.geocities.com/complementarytheory/MonadCK.pdf

I think that the organic natural number can be used as a model that represents the process of cells differentiation, starting by no specifications (represented by superposition) and ended by a specific structure and function for each cell of some organism.

For further reading, please look at http://www.geocities.com/complementarytheory/TOUM.pdf

Part 5:

Please be aware that there is a complementary relationship between NXOR logic and XOR logic.

Without it the concept of "many …" does not exist, and there is nothing beyond One.

Here is some analogy:

On a table there is a finite unknown quantity of identical beads > 1
and we have:

A) To find their sum.

B) To be able to identify each bead.

Limitation: we are not allowed to use our memory after we count a bead.

By trying to find the total quantity of the beads (representing locality) without using our memory (representing nonlocality) we find ourselves stuck in 1, so we need an association between nonlocality and locality if we wish to be able to find the bead's sum.

Let's cancel our limitation, so now we know how many beads we have, for example, value 3.

Let us try to identify each bead, but they are identical, so we will identify each of them by its place on the table.

But this is an unstable solution, because if someone takes the beads, put them between his hands, shakes them and put them back on the table, we have lost track of the beads identity.

Each identical bead can be the bead that was identified by us before it was mixed with the other beads.

We shall represent this situation by:

((a XOR b XOR c),(a XOR b XOR c),(a XOR b XOR c))

By notating a bead as 'c' we get:

((a XOR b),(a XOR b),c)

and by notating a bead as 'b' we get:

(a,b,c)

We satisfy condition B but through this research we define a universe, which exists between nonlocality and locality and can be systematically explored and be used to make Math.

What I have found through this simple cognition's_basic_ability_test is that ZF or Peano axioms "leap" straight to the "end of the story" where cardinal and ordinal properties are well-known, and because of this "leap" Infinitely many information forms that have infinitely many information clarity degrees (where each one of them is both global/local entity of this universe) are simply ignored and not used as first-order information forms of Math language.

In my opinion, any language (including Math) is first of all an information system, which means that fundamental properties like redundancy and uncertainty MUST be taken as first-order properties, but because our cognition's abilities were not examined when ZF or Peano axioms were defined, both redundancy and uncertainty were not included as first-order properties in our logical reasoning or in our fundamental axiomatic systems.

Also in my opinion, through this simple test we get the insight that any mathematical concept is first of all the result of cognition/object (abstract or non-abstract) interactions, that are translated to NXOR\XOR logic.




Sets and Multisets (some dialog):

My system can be defined as the complementary relations between Sets and Multisets.

From this point of view, each Natural number has several symmtrical states, where a multiset is the highest symmetry (a superposition), a set is the lowest symmetry (a broken-symmetry), and the rest states are combinations of symmetry and broken-symmetry, for example:

1
{x}

2
{{x}{x}} (superposition)
{{x}x} (broken symmetry)

3
{{x}{x}{x}} (superposition)
{{{x}{x}}x}
{{{x}x}x} (broken symmetry)

4
{{x}{x}{x}{x}} (superposition)
{{{x}{x}}{x}{x}}
{{{x}x}{x}{x}}
{{{x}{x}}{{x}{x}}}
{{{x}x}{{x}{x}}}
{{{x}x}{{x}x}}
{{{x}{x}{x}}x}
{{{x}{x}}x}x}
{{{x}x}x}x} (broken symmetry)

5
{{x}{x}{x}{x}{x}} (superposition)
{{{x}{x}}{x}{x}{x}}
{{{x}x}{x}{x}{x}}
{{{x}{x}}{{x}{x}}{x}}
{{{x}x}{{x}{x}}{x}}
{{{x}x}{{x}x}{x}}
{{{x}{x}{x}}{x}{x}}
{{{{x}{x}}x}{x}{x}}
{{{{x}x}x}{x}{x}}
{{{x}{x}{x}}{{x}{x}}}
{{{x}{x}{x}}{{x}x}}
{{{{x}{x}}x}{{x}{x}}}
{{{{x}{x}}x}{{x}x}}}
{{{{x}x}x}{{x}{x}}}
{{{{x}x}x}{{x}x}}
{{x}{x}{x}{x}}x}
{{{x}{x}}{x}{x}}x}
{{{x}x}{x}{x}}x}
{{{x}{x}}{{x}{x}}}x}
{{{x}x}{{x}{x}}}x}
{{{x}x}{{x}x}}x}
{{{x}{x}{x}}x}x}
{{{x}{x}}x}x}x}
{{{x}x}x}x}x} (broken symmetry)


...





I don't understand exactly how you're thinking about this stuff. However, I do know that if I count the number of different multisets of each "size" (with certain conditions, as I described in an earlier post), I get the same numbers as you get when you count whatever it is you're counting. So, in some sense at least, you are essentially talking about multisets of a certain kind. And I showed how to define multisets in terms of ordinary sets. That's what I meant when I said "but I did!".
If you use cardinality (of multisets or sets) as an order method, you are based only on a serial broken-symmetry.


To take a specific example, suppose we want to represent the multiset which contains 0 once and 3 twice. We can't represent it directly, as the set {0, 3, 3}, because this set is the same as {0, 3}. A set does not allow repetition of elements. Either it contains an element or it doesn't; for each possible element, those are the only two choices. But we can represent the multiset as a function f, where f(0) = 1 and f(3) = 2:


{ (0,1), (3,2) } =
{ {{0},{0,1}}, {{3},{3,2}} } =
{ {{ {} }, { {}, {{}} }}, {{ {{},{{}},{{},{{}}}} }, { {{},{{}},{{},{{}}}}, {{},{{}}} }} }

Obviously, this is hard to read, and there's no reason we can't extend the usual set notation to multisets, and write it as the easier-to-read "{0, 3, 3}" if we want to. But there's nothing fundamentally new in it. We don't need anything not in ZF to deal with it.

Let us check again the above very carefully.

1. A function gets an input and returns an output.

2. In this case the input cannot be related to a multiset, otherwise
we define a multiset by using a multiset, which is a circular definition.

3. So, the input must be several and different sets (if they are not different, we actually have a multiset as an input, which is a circular definition).

4. Any information that we get in the input is not lost in the output. So there is no problem to know what was the source of each member of {0,3,3} and this knowledge is used in order to identify each 3 in {0,3,3}, and let us use your own example to show it:


But we can represent the multiset as a function f, where f(0) = 1 and f(3) = 2:

{ (0,1), (3,2) } =
{ {{0},{0,1}}, {{3},{3,2}} } =
{ {{ {} }, { {}, {{}} }}, {{ {{},{{}},{{},{{}}}} }, { {{},{{}},{{},{{}}}}, {{},{{}}} }} }

Your functions have to visit each subset of { {{0},{0,1}}, {{3},{3,2}} } in order to get the data that satisfies the content of {0,3,3}.

In that case
if
visit a=0 of {0} of { {{0},{0,1}}, {{3},{3,2}} } ,
visit b=3 of {3} of {{3},{3,2}} of { {{0},{0,1}}, {{3},{3,2}} } ,
visit c=3 of {3,2} of {{3},{3,2}} of { {{0},{0,1}}, {{3},{3,2}} }
then
our output is not less than {0a,3b,3c} which is definitely not the multiset {0,3,3}.

5. If we say that knowledge that we have in the input does not fully exist in the output, then the input and the output are actually disjoint and we cannot conclude that the input is the building-block of the output.

6. If you say that your function is parallel and not a step-by-step, then you actually use a multiset in order to define a multiset (which is a circular definition).

7. Conclusion:

A set and a multiset cannot be defined by each other and they are mutually independent like two axioms.

8. There is a new mathematical universe that exists between sets and nultisets, that cannot be defined by using only ZF or ZFC.


The building-blocks of both of them are locality(XOR products) and non-locality(NXOR products).

Part 6:

I use Symmetry as the finest measurement tool of NXOR\XOR logic.

I showed in Part 5 that a multiset (a symmetrical structure of superpositions) cannot be defined by a set (a non-symmetrical structure of distinct identities).

They are mutually independent like two axioms, and both of them are NXOR\XOR products.

An NXOR product is exactly a non-local ur-element. Given any domain, the NXOR product is true only if it is the same inside NXOR outside of the given domain.

An XOR product is exactly a local ur-element. Given any domain, the XOR product is true only if it is not the same inside XOR outside of the given domain.

As for ur-element, please look at http://en.wikipedia.org/wiki/Urelement .

A consciousness is not less than a NXOR\XOR product, and it is the simultaneous bridging between non-locality(NXOR) and locality(XOR).

Bridging is relations between mutually independent ur-elements (a non-local ur-element(NXOR product) , a local ur-element(XOR product)).

Here is some illustration, based on a prism, which demonstrates the two extreme states of such a bridging:

http://www.geocities.com/complementarytheory/icmfig3.jpg

As can be seen, Uncertainty and Redundancy are first-order properties of it.

Given any finite cardinal NXOR\XOR logic systematically enables to define all NXOR\XOR products under it.

Furthermore, it explains the connection between complexity and consciousness's evolution in this NXOR\XOR illustration of what I call Cybernetic Kernels:

http://www.geocities.com/complementarytheory/MonadCK.pdf

Cybernetic Kernels are used in my cosmological model in:

http://www.geocities.com/complementarytheory/LPD.pdf

http://www.geocities.com/complementarytheory/Eventors.pdf

You can also look at an old paper of mine in:

http://www.geocities.com/complementarytheory/TAP.pdf

Here is some part of it:

Penrose tiling and number 5

http://www.geocities.com/complementarytheory/Penrose.jpg

As can be seen in the diagram above, the most top-left pattern is based on 5 cubes that cannot fully distinguished from each other, because they are identical and also preventing/defining each other by having common parts.

This pattern is a good analogy of QM wave/particle duality that can be found in some quantum element, before we measure it.

This example can be generalized to any finite cardinal.



Are you willing to suggest then that superposition must be a feature of some structure within the brain?

Superposition is an inseparable part of NXOR\XOR logic, where Platonic and non-Platonic realms preventing\defining their middle domain, and our realm is exactly a NXOR\XOR logic realm where each one of us is both its observer and participator.

As I understand it, the existence of a "pure" observer is a wishful thinking, or in other words, we are responsible for our actions\reactions as participators of this realm.

Please read http://www.geocities.com/complementarytheory/GEProject.pdf in order to understand my motive.

Could you briefly summarise the main point of the 6 posts above?

Thankyou.

There is no reasoning without NXOR\XOR relations.

Avoid it, and you cannot define anything beyond cardinal 1. Ignore it and you use a hidden assumption in order to get cardinal beyond 1.

dude...what is this ***t?

I think this is very interesting...

Am i just boring, or smth?

welcome

I think this is very interesting...

oh really...well than make sense out of it.

Have you never heard any of these concepts draqon, or even the terminology...?

Have you never heard any of these concepts draqon, or even the terminology...?

I have heard of some terminology...but I think this individual is nothing more than bogus.

I agree, this work is very obtuse... he could use defining and clearing it up more.

Other than that, his matrices would work well, giving the correct guidance.

How are you today?

I agree, this work is very obtuse... he could use defining and clearing it up more.

Other than that, his matrices would work well, giving the correct guidance.

How are you today?

Im good...been going here and there, my bike has gone flat. Monday will be very importan date for me, many things have to be done by than. And you?

Fine.

I'm actually dying today of a major hangover. I haven't had one like this in years.

Fine.

I'm actually dying today of a major hangover. I haven't had one like this in years.

hangover? huh...like marijuana/alcohol induced? or sleep deprivation?

Both the hash and the drink... please feel sorry for me ...

Both the hash and the drink... please feel sorry for me ...

I'll feel dead for you.

I will die for you

I will steal for you

I will do time for you

I will wait for you

I will kill you.

if I'll believe in you (http://www.youtube.com/watch?v=weNcDNTxqRY)

I will die for you.

Awe... you're so kind... :)

I think it is starting to work already!!!

Awe... you're so kind... :)

I think it is starting to work already!!!

I will pray for you



I will sell my soul for something pure,


someone like you

Now you're just chatting me up you flirt :)

lol