First a word about the Tortoise race, the key to understanding is that for any practical physical measurement Achilles and the Tortoise have stopped moving, which is the only way that Achilles does not pass the Tortoise.

Next it should be said that infinitesimals only occur as abstractions. Infinite terms can be compared to each other but not to finite terms.

For every paradox there is a false assumption. Examples the Tortoise continues to run, and the harmonic oscillator continues to vibrate in higher modes, both of which are not physically true, although unreal abstractions can be made for them.

I've learned that people who dwell on paradox are happy with an unsolved riddle, and disappointed to learn there is an answer.

Dear Mr. Jerry Decker ，

Thank you very much for your kindness and frank opinions, but what is your idea whether we are really able to produce infinite items each bigger than 1/2, or 100, or 1000000, or 1000000000000000000,…by that “Brackets-Placing Artifact” with limit theory from Un--->0 items in Harmonic Series and change Harmonic Series into an infinite series with infinite items each bigger than any positive constants (such as 100000000000000000000000000000)?

I will up load my new paper soon after it is published one month later and answer all of your questions ( two papers of On the Quantitative Cognitions to “Infinite Things” (I) and (II) have been up loaded).

Thank you again!

Sincerely yours,

Geng

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

---------“Fourteen Challenging Conundrums of Infinite Things’ Quantitative Cognitions” Produced by the Absence of the “Infinite Carriers’ Theory” and the Confusion of “Potential Infinite” and “Actual Infinite”

OUYANG Geng

(Department of Mathematics, Minnan Normal University, Zhangzhou, Fujian, China, 363000)

Abstract: The fundamental fatal defects on “infinite things’ quantitative cognitions” since antiquity in present classical infinite related science system disclosed by the suspended typical “’infinities—infinitesimals’ paradox families” are analyzed and studied, “fourteen challenging conundrums of ’infinities—infinitesimals’ quantitative cognitions” in modern mathematical analysis and set theory are raised. Conclusion: the absence of the “infinite carriers’ theory” and the confusion of “potential infinite” and “actual infinite” in present classical infinite related science system has been unable people to answer such a basic question self-justificationly since antiquity-------- are “infinite related number forms” (numerical related infinite things) “potential infinite” or “actual infinite”? If they are “actual infinite”, how to cognize and manifest them quantitatively? If they are “potential infinite”, how to cognize and manifest them quantitatively? The fundamental defects have been formed an unconquerable “infinite things’ quantitative cognizing obstacle”. Discarding the contradictory and unscientific “potential infinite--actual infinite” theory, studying and developing systematically the “abstract infinite concept-- abstract infinite concept carrier” related new infinite theory system is the first step we should take to solve those fundamental errors and defects, and to dispel the thousands years old suspended huge black cloud of “’infinities—infinitesimals’ paradox families” over the sky of present classical infinite related science system.

Key words: Philosophy of mathematics; The foundation of mathematics; The absence of the “infinite carriers’ theory”; The confusion of “potential infinite” and “actual infinite”; The14 challenging conundrums of “infinite things” quantitative cognitions; The infinite related paradox families; New infinite theory system

My studies proved that the essence of Zeno’s Achilles--Turtle Race Paradox is making good use of unaware fundamental defects of the “absence of infinite carriers theory” and the confusion of “potential infinite--actual infinite”: (1), with the idea and operation of “potential infinite” to make people believe that if the Turtle is unceasingly crawling forward, there will be endless distance composed by the tiny distance (a kind of “infinite” related number forms) from Turtle’s steps for Achilles to cover; (2), with the idea and operation of “actual infinite (including present classical limit theory)” to make people believe unconsciously that no matter how big Achilles’ steps are, how fast he runs (even with the speed of modern jet plane), how long the time for him to run and how tiny distance the Turtle produces, there sure be endless distance composed by the tiny distance from Turtle’s moving for Achilles to cover; (3) constructing paradoxes or “proving” that Achilles will never catch up with the Turtle. These paradox family members, though from different eras, disclosed deeply a fact that in present classical infinite related mathematical analysis, non of our quantitative cognizing ideas and activities to “infinite small, infinite few, infinite great, infinite many” can overstep two quantitative cognizing obstacles inevitably laid down by the fatal fundamental defects in present classical infinite related science and mathematics: (1), unable to understand, cognize scientifically (self-justification) what the mathematical contents and number forms of Un 0 and Vn ∞ are; (2), abuse quantitative cognizing tools for “infinite things” (such as “limit theory”)

As the members from the same paradox family generated by the same classical “potential infinite--actual infinite” based science and mathematics theory system, the role of Great Achilles in Zeno’s Achilles--Turtle Race Paradox 2500 yeas ago is exactly the same thing by the title of “Brackets-Placing Artifact” good at manufacturing infinite items each bigger than 1/2, or 100, or 1000000, or 100000000000000000000000000000,…from Un--->0 items in Harmonic Series. We have to admit a truth: 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 based on modern limit theory " has been confirmed as a truth and a unimpeachable basic theory in our science (mathematics) means the suspended Ancient Zeno’s challenging statement of “not matter how fast Achilles runs, he will never catch up with the slowly crawling Turtle in Achilles--Turtle Race” has been confirmed with exactly the same operations and same ideas basing on modern limit theory as a “strict mathematical proven” truth and a unimpeachable theorem------it would be Great Zeno’s Achilles--Turtle Race Theorem but not Suspended Zeno’s Achilles--Turtle Race Paradox!