（1）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”.

（2）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.

（3）The above different operating ideas and results of quantitative studying and cognizing on “elements in infinite sets” in present classical infinite set theory are affirmed scientific and both aceptable. The one-to-one coresponding operation idea and result of quantitative studying and cognizing on Rational Numbers Set and Natural Numbers Set is the most typical example. In present classical infinite set theory, on the one hand, we can prove the general acepted conclution of “Rational Numbers Set has exactly same elements as Natural Numbers Set, so it is countable” by the above first operating idea and result; on the other hand, we can also prove “Rational Numbers Set has infinite more elements than Natural Numbers Set, so it is uncountable” by the above second operating idea and result, because just a tiny portion of rational numbers from Rational Number Set (such as 1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) can map and use up (bijective) all the numbers in Natural Number Set. This is the newly discoved “’Countable-uncountable’ Paradox of Rational Number Set”. The conclution is: the quantitative studying and cognizing theories and operations in present classical infinite set theory are lack of scientificity.

It is because of the fundermental defects of the absence of “infinite carrier theory” and the unscientific concepts of “potential infinite--actual infinite” in present classical infinite set theory that have been making people unable to know which of the above threedifferent operating ideas and results is scientific and why, unable to know scientifically and systematically the relationship between “elements in infinite set” and “infinite set” , unable to know scientifically and systematically the “nature, existing condition, manifestation and relationship among each other” of sets through their elements, ... This situation has been making people unable to know clearly since antiquity how to carry on quantitative studying and cognizing on elements in infinite set scientifically and systematically, resulting in the production and suspending of all kinds of “infinite things’ quantitative cognizing paradox families” in present classical infinite set theory.