In Standard and Non-Standard Real analysis expressions like 0.111…[base 2] = 0.222…[base 3] = … = 0.999…[base 10] = * 1* , where only

*is a number and the rest expressions are no more than different representations of it.*

**1**Both Standard and Non-Standard Real analysis are based only on local mathematical objects, and as a result the place value method is considered as a set of sequences of numerals, where each string of numerals is no more than a different representation of the same **R** member, and only **R** member is considered as a number.

If the Real-line is a NXOR outcome, then it is a non-local ur-element, and it is not a set of infinitely many distinct members.

As a result **R** set is an incomplete mathematical object, if it is compared to the non-local ur-element. ( http://en.wikipedia.org/wiki/Urelement )

Furthermore, each **R** member is a mathematical object that has an exact location along the non-local ur-element Real-line, but between any given pair of local members there are non-local mathematical objects that their exact location along the non-local ur-element Real-line cannot be determinate.

These mathematical objects are called non-local numbers, and a particular case of them can be represented by the known place value method.

Let us examine the relations between the non-local **number** * 0.111…[base 2]* and the local

**number**

*.*

**1**As can be seen, the non-local **number** * 0.111…[base 2]* does not have a zenith of infinitely many zeros as the local

**number**

*has, and as a result*

**1****number**

*has an exact location along the non-local ur-element Real-line, and*

**1****number**

*does not have an exact location along the non-local ur-element Real-line.*

**0.111…[base 2]**We can ask, what number exists between **number** * 0.111…[base 2]* and

**number**

*?*

**1**The answer is: a non-local number that its base > 2, for example **number** **0.222…[base 3]:**

Some claims that if (for example) * 0.999…[base 10]* <

*than there must be*

**1***, but this is an invalid expression because there cannot be infinitely many zeros and then 1, which contradicts the existence a non-finite sequence of zeros.*

**0.000…1[base 10]**My answer is: If the Real-line is a non-local ur-element, then an expression like * 0.000…1[base 10]* means that its exact location along the non-local ur-element Real-line cannot be determinate, and “…1” is the exact notation that rigorously defines this indetermination.

Furthermore, * 0.000…1[base 2]* >

*> … >*

**0.000…1[base 3]***> … and each one of them is a non-local number.*

**0.000…1[base n]**

———————————————————————————————————————————————————

Let us examine how the concept of the Real-line is understood from the NXOR\XOR point of view.

By Standard Logic (where non-locality is ignored and NXOR is used as a hidden assumption) the real-line is a collection\sequence of infinitely many distinct members, where each member is a XOR product (what I call a local member, which can be **in** XOR **out** of some given domain that is notated by “{” and “}” ).

The truth table of XOR (if Membership is examined) 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

As a result, the real-line is no more than a collection\sequence of infinitely many local members.

From a NXOR\XOR logic the power of the real-line is determined by NXOR.

The truth table of NXOR (if Membership is examined) is:

**in** **out**

0 0 → T (**in** , **out** are the same) = { }

0 1 → F

1 0 → F

1 1 → T (**in** , **out** are the same) = {* **}** *

As a result, the real-line is not less than a non-local ur-element.

In order to understand what are the relations between locality and non-locality from a NXOR\XOR logic (where NXOR product of non-locality is not ignored, and we avoid the hidden assumption) please let’s examine the Riemann sphere model from a NXOR\XOR point of view.

Let . be the minimal representation of locality (a XOR product)

Let __ be the minimal representation of non-locality (a NXOR product)

Let *Z* be a NXOR\XOR product.

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) = {* **}** *

Let us examined the Riemann sphere model( http://en.wikipedia.org/wiki/Riemann_sphere ) , where . And __ are not metric-space representations, but they are representations of NXOR ({* **}** * ) and XOR ({_} that can be reduced to {.} representation):

By the section above, we can clearly see that if the non-finite is not less than a non-local ur-element, than any non-finite collection\sequence is no more than an incomplete mathematical object (when compared to the non-local ur-element) because no non-finite collection\sequence can be logically a non-local ur-element (in this model each local member ({.}) is an intersection, and the non-finite itself ( the non-local ur-element that is represented by{* **}** * ) does not have any intersections (it is an atom)).

The same thing holds in the sports car analogy:

At a non-finite speed this sports car is exactly the non-local ur-element (notated as X-axis) and infinitely many cars cannot reach a non-finite speed (in a non-finite speed, our observed object cannot be but a non-local ur-element).

Also if we are exactly at point 0, than we do not have any sports car, so no sports car can reach point 0 and still be considered as a member of the non-finite collection of sports cars.

So in both cases ( the local case that is represented by . , or the non-local case that is represented by __ ) any non-finite collection is **logically** incomplete.

For further reading please look at http://www.geocities.com/complementarytheory/Paradigm-Shift.pdf and http://www.geocities.com/complementarytheory/TOUM.pdf .