Basing on “potential infinite--actual infinite”, do we have only following three different operating ideas and results of quantitative studying and cognizing on “elements in infinite sets” in present classical infinite set theory?

（1）With the idea of “potential infinite”: Denying the essential differences of infinite set elements’ “special nature, special existing condition, special manifestation and special relationship among each other”, so, elements in all different infinite sets are only heaps of infinite “indiscriminative, nonquantitative, abstract points” and it is unnecessary and impossibale to have any quantitative studying and cognizing on “infinite elements (points) in infinite sets”, the number of elements in all infinite sets are the same and it is just infinite. For example, elements in many infinite mother sets and their sub-sets are in fact all the same stuff of infinite “indiscriminative, nonquantitative, abstract points” without any differences of “nature, existing condition, manifestation and relationship among each other”. The typical cases are “the element numbers in Rational Numbers Set and Natural Numbers Set are equal, the element numbers in Natural Numbers Set and Even Numbers Set are equal and they are just all infinite”. The conclution is: any infinite set has limitless, endlees, infinite elements, so their “one-to-one coresponding operations” of quantitative studying and cognizing on “infinite elements (points) in infinite sets” can be carried on for ever and their element numbers are all “equally infinite”. Going along this train of thought, one can understand and construct those theorms and proofs related to the idea of “different infinite sets, equal element numbers”.

（2）With the idea of “actual infinite”: Affirming the essential differences of infinite set elements’ “special nature, special existing condition, special manifestation and special relationship among each other”, so, elements in all different infinite sets can be “discriminative, quantitative visible and tangible mathematical things” and it is necessary and possibale to have all kinds of quantitative studying and cognizing on “infinite elements in infinite sets”. For example, there are different “special nature, special existing condition, special manifestation and special relationship among each other” betwee elements in Real Numbers Set and Natural Numbers Set, so, these two infinite sets have different element numbers. The typical cases are “the element numbers in Real Numbers Set are more than that of Natural Numbers Set, the element numbers in Power Set are more than that of its original set and they are all inequal”. The conclution is: Different infinite sets may have different element numbers. So, in the “one-to-one coresponding operations” of quantitative studying and cognizing on infinite elements of two different infinite sets, the elements in smaller set with fewer elements are consumed and finished soon, it means the element numbers in such an infinite set are not endlees and limitless at all (fake infinite); but in the begger set, infinite many elements are left during this operations, this means its elements are endlees and limitless (real infinite), never be consumed and finished at all, the “one-to-one coresponding operations” here can never be carried on for ever at all. Going along this train of thought, one can understand and construct those theorms and proofs related to the idea of “different infinite sets, different element numbers”.

（3）With the idea of mixture “potential infinite--actual infinite”: The above different operating ideas and results of quantitative studying and cognizing on “elements in infinite sets” in present classical infinite set theory are both acknowledged “OK” and accepted. This inevitably makes us unable to cognize scientifically and clearly the infinite sets and their elements we are facing to in many situations------- being caught in a paradoxical status of arbitrary, self-contradictory and non-self-justification. So, Russell's Paradox as well as its family members in different periods of time have been generated, developed and suspended. Following two cases are typical examples--------First case: with the idea of “potential infinite”, we claim “certain infinite set with its defined elements can be constructed”, than we claim “some of these elements actually can not be included within this very infinite set” with the idea of “actual infinite” and find out those “belonging to but uncontained elements of this set” to prove all kinds of family members of Russell's Paradox in different periods of time (My stuedies proved that whenever we come to any case of “elements belonging to but impossibly contained in a set, a defined group of things”, we come to a situation of “confusing whole and portion” -------- traped immediately into “potential infinite--actual infinite”. Liar’s Paradox, Post Card Paradox,... are all family members of Russell's Paradox) . Second case: with the idea of “potential infinite”, we have proved “the elements in Rational Number Set are equal to the elements in Natural Number Set”, but at the same time we can prove “the elements in Rational Number Set are infinite more than the elements in Natural Number Set” because just a tiny portionof the rational numbers from Rational Number Set (such as a subset of 1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) can map and use up (bijective) well all the numbers in Natural Number Set (1,2,3,...,n,...), so a lot of (infinite) rational numbers are left behind in this one-to-one mapping from the Rational Number Set onto the Natural Number Set and construct the “’Countable-uncountable’ Paradox of Rational Number Set”. Going along this train of thought, one can understand and construct all kinds of family members of Russell's Paradox in different periods of time. The conclution is: the quantitative studying and cognizing theories and operations in present classical infinite set theory are too arbitrary and lack of scientific basic theory.