End-Point Paradox ( Infinity Paradoxes rise again

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The (End Point) Paradox

Note: this is NOT Zeno`s dichotomy Paradox, in fact this is a more
general paradox, and unlike Zeno`s dichotomy Paradox, It cannot be
solved by calculus, infinite series or any other method used to solve
zeno paradoxes, however, any solution for it should be a solution also
for all Zeno paradoxes. this paradox makes a clear mathematical
problem out of the supertask issue which is used to be seen as a
philosophical dilemma more than a mthematical problem.


(Figure 1)


P0<----------------- 1=0.999... ------------------->P1
___I______________________________._________._____.__I____________
..k1 k2 k3

The real number 1 has one start point P0 and one end point P1 on
the real numbers line
------ ------
The infinite series ( 0.999...= 0.9 + 0.09 + 0.009 +...) has a first term
but has no final term ---> so, it has a start point P0 but cannot have
an end point because there is no final term to set that end point
------ -------
Because 1=0.999... , Then ---> the end point that bound the real
number 1 must be also an end point for the series 0.999...
------ -------
Conclusion: The end point P1 cannot exist and must exsist at the same
time!.

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Here is the Paradox in more Detalis:

1) The least upper bound axiom LUB define the real number 1 to be the
least upper bound of {0.9, 0.99, 0.999, 0.9999,...} so, 0.999...=1

2) In 0.999... there is no final 9 because such assumption will
make 0.999... < 1 and violates the LUB axiom.

3) Every 9 in 0.999... represents a positive real number Ki, where
{ Ki belongs to R , K>0 , i belongs to N}
For example K1=0.9 , K2=0.09, K3=0.009, K4=0.0009,.... And so
on
..... + So, 0.999... = K1 + K2 + K3 + K4 + K5

4) In ( 0.999... ) there are infinite positive real numbers Ki

5) Every positive real number Ki that represents a term in the
infinite series 0.999... represents also a distance on the real numbers
line.

6) Every distance represented by any Ki is a line segment bounded by
two fixed mathematical points at the beginning and at the end.

7) The real number 1 which is represented by the series 0.999... is
bounded by two fixed mathematical points P0 and P1 on the real numbers
line (Figure 1)

8) Because 1=0.999... then those two fixed points that bound the number
1 on the real numbers line should also bound the infinite sum 0.9 +
0.09 + 0.009 + ...


9)The fixed point P0 is the start point of the segment line that
represent the number 1 , and at the same time it is the start point of
the first term in the infinite series 0.999... which is K1= 0.9

10) The end point P1 is the end point of the segment line that
represent the real number 1, hence it must be also the end point of
the line segment that represents the infinite series 0.999

11) Because the end point P1 is the end point of the line segment
0.999... , then it is also the end point for one and only one partial
line segment Ki that makes with the others the whole line segment
0.999... we call this partial line segment Kz ( any point on the real
numbers line cannot represent more than one real number).

12) There cannot be any more positive real numbers (partial line
segments) Ki added after the term Kz because adding any more
positive real number Ki will shift the end point P1 which is the end
point of the number 1 and as a result it makes 0.999... >1 which is
not allowed.

13) Because there cannot be any term after Kz, then it is a final
term in the infinite series ( 0.9 + 0.09 + 0.009 + ... )

14) It is a contradiction, because the existence of a final term in
an infinite series contradicts the definition of the infinite series
itself (there cannot be a final term in any infinite series because
it makes it finite!); or worse, it assumes reaching a very small
positive real number Kz>0 and at the same time there cannot be any
other real number between Kz and 0 , which violates the axioms of
real numbers ( Least Upper Bound).

So, to declare the problem briefly:

1- There cannot be a specific end-point for any real number
represented on the real number line because this contradicts the
definition of infinite series and the axioms of real numbers (least
upper bound).

2- If there is no specific end-point for any real number on the real
numbers line, then this makes all real numbers equal to each other and
to which is a contradiction too. .



Made by: Said.H.Alyami
E-mail: said_yam@xxxxxxxxx

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