**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 measure**s** 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).

**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.