Infinite with its not-knowing-what “potential infinite--actual infinite” are right there in our science ever since and none of the “infinite things” in present classical infinite related science system can run away from such defected fundamental “potential infinite--actual infinite”.
The suspended “Ancient Zeno’s Achilles--Turtle Race Paradox” and its modern version of the newly discovered Harmonic Series Paradox disclose from the “infinitesimals” aspect the unavoidable confusing defects of “potential infinite and actual infinite”.
The “transfinite number related troubles” disclose from the “infinities” aspect the unavoidable confusing defects of “potential infinite and actual infinite”.
Looking back into our infinite related science history, we can see many different “infinite theories”; such as “infinite is a kind of psychological thing”, even each person can have his/her own “infinite theory system”, so there can be many different definitions for the concept of “infinite” in science. Some people think “infinite theory” is only for infinities and not for infinitesimals, even deny the existence of infinitesimals (Cantor is a typical representative for this idea). But most popular, we have “potential infinite--actual infinite” theory as main stream ever since which declares all the infinite things can be divided into either “potential infinite” or “actual infinite” and further construct many infinite ideas and concepts: supper potential or actual infinite, supper supper potential or actual infinite, supper supper supper potential or actual infinite, …; Infinite Set A is more infinite than Infinite Set B, more more infinite than Infinite Set B, more more more infinite than Infinite Set B, more more more more infinite than Infinite Set B, …; bigger potential or actual infinite--smaller potential or actual infinite, …; more potential or actual infinite--less potential or actual infinite, …; although the heated debates on “what are potential infinite and actual infinite” since antiquity have been being endless.
From the “infinitesimals” aspect, the suspended “Ancient Zeno’s Achilles--Turtle Race Paradox” and its modern version of the newly discovered Harmonic Series Paradox disclose the unavoidable confusing defects of “potential infinite and actual infinite”.
From the “infinities” aspect, the “transfinite number related troubles” disclose the unavoidable confusing defects of “potential infinite and actual infinite”.
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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