Many people believe that it is for sure that Achilles can catch up with a turtle-------a hungry lion is for sure to catch up with a running man.

Dear Mr. Mohamed El Naschie, thank you for your view point.

Could I ask 2 questions?

1, Is “CARDINALITY” anything to do with infinitesimals?

2, Should infinite theory cover both theories of infinity (infinitebigs) and theories of infinitesimals?

Best regards,

 Geng

If we forget about physics and concept of time, then the problem is in affiliation of asymptotic point (Turtle) to continuum in which Achilles is catching up Turtle. In numerological world of causal relationships is impossible to reach this point because the concept of a continuum (essentially abstract) breaks down. In Newton's time which is actualized to continuum all a bit wrong.

Geng, it's a little reflection upon our productive dialogue about potential and actual infinity and works of Amrit.

Dear Mr. Amrit Sorli and Mr. Vasiliy Fedorovich Komarov, thank you very much for your frank ideas.

Would you please give your frank idea on following “strict mathematical proven” modern version of Ancient Zeno’s Achilles--Turtle Race Paradox, which can be found in many current higher mathematical books written in all kinds of languages:

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                       （1）

=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．      （２）

>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．               （3）

=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                                （4）

Each operation in this proof is really unassailable within present science theory system. But, it is right with present modern limit theory and technology applied in this proof that we meet a “strict mathematical proven” modern version of Ancient Zeno’s Achilles--Turtle Race Paradox: the “brackets-placing rule" decided by limit theory in this proof corresponds to Achilles in Zeno’s Paradox and the infinite items in Harmonic Series corresponds to those steps of the tortoise in the Paradox. So, not matter how fast Achilles can run and how long the distance Achilles has run in “Achilles--Turtle Race”, there would be infinite Turtle steps awaiting for Achilles to chase and endless distance for him to cover, so it is of cause impossible for Achilles to catch up with the Turtle; while in this acknowledged modern divergent proof of Harmonic Series, not matter how big the number will be gained by the “brackets-placing rule" (such as Un’ >10000000000) and how many items in Harmonic Series are consumed in the number getting process by the “brackets-placing rule", there will still be infinite Un--->0 items in Harmonic Series awaiting for the “brackets-placing rule" to produce infinite items each bigger than any positive constants, so people can really produce infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 10000000000), “strictly proving” that Harmonic Series is divergent. In so doing, the conclusion of “infinite numbers each bigger than any positive constants” can be produced from the Un--->0 items in Harmonic Series by brackets-placing rule and Harmonic Series is divergent" has been confirmed as a truth and a unimpeachable basic theory in our science (mathematics) while “the statement of Achilles will never catch up with the Turtle in the race” in Ancient Zeno’s Achilles--Turtle Race Paradox has been confirmed as a “strict mathematical proven” truth and a unimpeachable theorem------it would be Great Zeno’s Theorem but not Suspended Zeno’s Paradox! ?

Thank you again!

Best regards,

Geng

Geng, with surprise for itself I discovered the connection of this problem with non-compact manifolds. This is something that concerns the physics of course.

I agree, perhaps in my thinking are mixed dichotomy paradox and Achilles - Turtle Race. Sorry for offtopic.

Dear Mr. Vasiliy Fedorovich Komarov, thank you.

According to my studies, "applied infinite" has a lot to do with physics--------there is no mathematics without physics.

Your researches may bring something new in our work on the infinite related foundation of mathematics.

Yours, 

Geng

Geng, there is another simple way of the graphical solving of the paradox. If the chart of steps complied with scaling (scale invariance) then Achilles did not overtake the Turtle. Paradox exists only in the discrete realm.

Dear Mr. Vasiliy Fedorovich Komarov, thank you.

Could that “simple way of the graphical solving of the paradox" work for the above newly discovered family member of 2500-year old Zeon’s Paradox------Harmonic Series Paradox?

Yours,

 Geng

Dear Geng, harmonic series involve some evaluation manipulations on the steps of Zeno characters. I will try whenever possible to consider them closely.

Dear Mohamed, you manipulate labels of terms without trying to explain them in simple words. Formal approach can not prove anything to experimenters. Your attachment to the formal terminology leaves out the understanding of physics not only for us. Sometimes I think that there is a sensible grain. Try to rethink your approach to showcase own ideas, they might be interested for us. And you should not mix in one pile all at once.

Dear Mr. Mohamed El Naschie and Mr. Vasiliy Fedorovich Komarov, thank you.

According to my studies, “potential infinite” and “actual infinite” have been two important concepts in our present classical infinite related foundation of science (mathematics) theory system at lest since Zeno’s time more than 2500 years ago. The defects in this field have produced “the 2500 years old black cloud of infinite related paradoxes over mathematics sky.”

1, When paying equal attention on “potential infinite” and “actual infinite”, we meet the uncompromised logical contradiction and the typical case is the “strict mathematical proven” ancient Zeno’s Paradoxes and modern Paradox of Harmonic Series.

2, When paying more attention on “potential infinite” and neglect “actual infinite”, we meet all the ideas and theories of “all infinite things are the same endless without any inborn and numerical natures” and the typical cases are all the ideas and theories of the countable infinite sets.

3, When paying more attention on “actual infinite” and neglect “potential infinite”, we meet all the ideas and theories of “all infinite things are not the same limitless and endless, they have inborn and numerical natures” and the typical cases are all the ideas and theories of power set and uncountable infinite sets (infinite Set R has more elements than infinite Set N): more infinite, more more infinite, more more more infinite,…; higher infinite, higher higher infinite, higher higher higher infinite,…; bigger infinite, bigger bigger infinite, bigger bigger bigger infinite,…; supper infinite, supper supper infinite, supper supper supper infinite,…; Continuum Hypothesis,…

Yours,

 Geng

 Dear Mr. Colin James III ， thank you.

It is ok that we may forget the “time” and “space”, but we have to solve practical mathematical problems such as “can we really produce infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 10000000000)---------- a strict mathematical proof of Zeno’s Achilles--Turtle Race Paradox?

Yours sincerely,

 Geng

Dear Mr. Amrit Sorli, thank you.

It has been proved that the newly discovered Harmonic Series Paradox is a modern mathematical version of the Achilles--Turtle Race Paradox and all the members of this paradox family can never be solved in present infinite related science theory system.

The newly discovered Harmonic Series Paradox is surely nothing to do with the “time” and “space”.

What do you think?

Yours sincerely,

 Geng

Five of my published papers have been up loaded onto RG to answer such questions:

1，On the Quantitative Cognitions to “Infinite Things” (I)

https://www.researchgate.net/publication/295912318_On_the_Quantitative_Cognitions_to_Infinite_Things_I

2，On the Quantitative Cognitions to “Infinite Things” (II)

https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II

3，On the Quantitative Cognitions to “Infinite Things” (III)

https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III

4 On the Quantitative Cognitions to “Infinite Things” (IV)

https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV

5 On the Quantitative Cognitions to “Infinite Things” (V)

https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V