Our science history tells that we human have been arguing and debating on the nature and definition of “potential infinite and actual infinite” for at least 2500 years since Zeno’ time, and there are still no results nowadays. So, when facing “infinite things” in present classical infinite related science theory system, no one is sure whether they are “potential infinite things” or “actual infinite things” and are not sure how to treat them scientifically------- many suspended old and new infinite related paradox families are produced (such as 2500-year old Zeno’s Achilles--Turtle Race Paradox and so many modern versions of Zeno’s Achilles--Turtle Race Paradox, Russell's Paradox and so many modern versions of Russell's Paradox, …)by the confusing of “potential infinite and actual infinite” and these paradox family members are surely unsolvable in present classical infinite related mathematics where they were produced and nourished.
One may say “look at the history of mathematics and don’t mind the arguing and debating on the nature and definition of potential infinite and actual infinite, our mathematics goes well without the definition of infinite, just close one eye and open another eye in the field of infinite and we are used to this since Zeno’ time”.
But the facts are: more and more suspended infinite related paradox families are produced by the “confusing of potential infinite and actual infinite” such as infinitesimal relating paradoxes in analysis, infinity relating paradoxes in set theory, both infinitesimal and infinity relating paradoxes in the ideas and skills of “Cantor’s diagonal-contradictory proofs and the conclusion on “Real Number Set has more elements than Natural Number Set (infinite elements in Real Number Set is more infinite than that in Natural Number Set-------Infinite R is more infinite than Infinite N)”.
Can we really just close one eye and open another eye in the field of infinite and force us be used to the fundamental defects in our mathematics?
Trying our best to have scientific foundation for our mathematics is a must we should shoulder sooner or later beyond our will.
Unsolved problems in mathematics are an incentive for the evolution of mathematics and other sciences. This is the conclusion of the history of mathematics (and other sciences).
Dear Professor Vladimir A. Kulchitsky , thank you for your insightful idea.
Yours,
Geng
Dear Dr. Chris Cook, thank you for your quantitative ideas on the “infinite”.
In fact, there are 3 problems make us unable to discuss “quantitative infinite” scientifically in our present “potential infinite-actual infinite” related classical science system:
1, no one knows whether the “infinite” mentioned in any “quantitative infinite” question “potential infinite” or “actual infinite” because no one and nothing can run away from the confusing of “potential infinite” and “actual infinite”;
2, even some one decides that the “infinite” is sure to be “potential infinite” or sure to be “actual infinite”, no one in the world now is able to define scientifically what “potential infinite” is and “actual infinite” is-------there have been no scientific definitions for both “potential infinite” and “actual infinite” at least since Zeno’s time 2500 years ago, so we have endless debates on them;
3, the “infinite” is “non- number number”--------- on the one hand, we have to cry out loudly and firmly first in mouth that “infinite small and infinite big are not numbers” while these “non-number infinite numbers” are participating all kinds of calculations with numbers; on the other hand, we tell in mind without any hesitation: “forget what have been just said in mouth for those very infinite small and infinite big, they are in fact numbers other wise no practical numerical operations can be done”. Most of us have to play the role of noble ministers in Andersen’s fairy tale The Emperor's New Clothes, compiling wonderful sayings to persuade everyone believe that “this emperor is a special one and he has no clothes on while has clothes on”--------the infinite small and infinite big participants in number calculations are “numbers while non-numbers”, “finite things while infinite things”, “potential infinite things while actual infinite things”,….
Yours,
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
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