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NXOR\XOR Real Analysis
 Posted: 28 August 2007 02:34 PM [ Ignore ]
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In Standard and Non-Standard Real analysis expressions like 0.111…[base 2] = 0.222…[base 3] = … = 0.999…[base 10] = 1 , where only 1 is a number and the rest expressions are no more than different representations of it.

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 1 has, and as a result number 1 has an exact location along the non-local ur-element Real-line, and number 0.111…[base 2] does not have an exact location along the non-local ur-element Real-line.

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] < 1 than there must be 0.000…1[base 10], 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.

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

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

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.

[ Edited: 15 February 2008 04:24 AM by DoronShadmi ]
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 Posted: 28 August 2007 04:09 PM [ Ignore ]   [ # 1 ]
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 Posted: 28 August 2007 09:05 PM [ Ignore ]   [ # 2 ]
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I’ll be part of the audience and let Alon and narwhol deal with this thread.

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