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:

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:

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:

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:

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:

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

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).

The Whole is a property that is not related separately to each set, but it is a common property of both of them. In the case of numbers, the Whole is called Size.

If NXOR product (the concept of Size in this case) is not a hidden assumption, then we do not mix between size’s representation and size itself (which is the number of members from both sets that are included or equal to this size).

Futhermore, any non-finite collection is incomplete by the organic paradigm and a non-finite collection is simply a collection that no one of its elements is its final element.

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.

For further reading please look at:

http://www.centerforinquiry.net/forums/viewthread/2893/P30/#20917

http://www.centerforinquiry.net/forums/viewthread/2893/P30/#20918

http://www.centerforinquiry.net/forums/viewthread/2893/P30/#20919

Thank you.