Thank you my dear Mr. downvoter, would you be so kind as to join the discussion and tell me your frank point of view what mistakes in the above question and how to correct the mistakes. That is what we really need in “question--discussion”-----not just hiding somewhere and killing.

I have known from very beginning more than 40 years ago that my idea and my work is against our current trend and I have suffered a lot in my life already------I can stand more downvotes.

Although my RG score is getting lower and lower week by week by the downvoters, I still warmly welcome opposite points of view in my “question--discussion” from which I benefit a lot.

Beg you and thank you again for your kind participation.

Yours,Geng

“Potential infinite” and “actual infinite” have been two important concepts in our present classical mathematical theory system at lest since Zeno’s time more than 2500 years ago.

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 endless, they have inborn and numerical natures” and the typical cases are all the ideas and theories of uncountable infinite sets: 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,…

Can we really have many different definitions with different natures for the concept of “infinite” in human science?

Can we name any infinite things “potential infinite” or “actual infinite” or more infinite or supper infinite or… as we like?

Can we pay attention on “potential infinite” and neglect “actual infinite” or vice-versa as we like?

…

That is why we need A Revolution in the Infinite Related Foundation of Mathematics to dispel “the huge black cloud of infinite related paradoxes over mathematics sky”.

Dear colleagues, anyone knows how to solve following problems:

1, According to the definition, the original Even Number Set and Odd Number Set is each half of the Natural Number Set (otherwise it can not be called Even Number Set and Odd Number Set). People have been using two deformed Natural Number Set of (1x2, 2x2, 3x2,...,nx2,...) and (1x2+1, 2x2+1, 3x2+1,...,nx2+1,...)  and just used the components of even numbers and odd numbers (natural numbers, but not the whole thing of“2,4,6,8…2n” and “1,3,5,7,9…nx2+1”) to map natural numbers, of cause they map very well because it is N ----> N itself but not 2n ----> N and nx2+1----> N.

What are the mappings to natural numbers: the even numbers and odd numbers or the components of even numbers and odd numbers?!

2,The elements of tiny portion of rational numbers from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …)  map and use up (bijective) all the numbers in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so infinite rational numbers (at least 2,3,4,5,6,…n,…) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set”------- Rational Number Set has infinite more elements than Natural Number Set and they are not bijective at all.

What decides the same cardinality of two sets?

In present classical infinite science system, people have been paying too much attention to the forms but too less to the ontology at less since Zeon’s time.

Regards,

Geng

Dear ProfessorWaldemar Koczkodaj,  how do you think of the following truth:

The elements of tiny portion of rational numbers from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) map and use up (bijective) all the numbers in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so infinite rational numbers (at least 2,3,4,5,6,…n,…) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set”------- Rational Number Set has infinite more elements than Natural Number Set.

Best,

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