As The First Generation of Infinite Set Theory is based on present classical infinite theory system, contradictory concepts of "potential infinite” and “actual infinite" make people unable to understand at all what the mathematical things being quantitative cognized in set theory are-------- are they "potential infinite things” or “actual infinite things " or the mixtures of both or none of both? People have been unable to understand at all what kind of relationship between the quantitative cognizing theory and the unavoidable concepts of "potential infinite, actual infinite" in set theory: If the mathematical things being quantitative cognized are "potential infinite”, what kind of "potential infinite” cognizing idea, operations and results should people have; if the mathematical things being quantitative cognized are "actual infinite”, what kind of "actual infinite” cognizing idea, operations and results should people have; if the mathematical things being quantitative cognized are the mixtures of both or none of both "potential infinite” and "actual infinite”, what kind of mixing cognizing idea, operations and results should people have? Are there "one to one correspondence" theories and operations for "potential infinite elements” or “actual infinite elements" or the mixtures of both or none of both? Why?
Therefore, it is very free and arbitrary for people to conduct quantitative cognitions to any infinite related mathematical things in The First Generation of Infinite Set Theory: It can either be proved that there are as many elements in Rational Number Set as there are in Natural Number Set or that there are more elements in Rational Number Set than that in Natural Number Set; the T = {x|x📷x}theory can either be used to create Russell’s Paradox or to create "Power Set Theorem", make up the story of “the Hilbert Hotel forever with available rooms” ------- strictly make all the family members of the Russell's Paradox mathematicization and turn all the family members of Russell's Paradox into all kinds of Russell's Theorems; ...
However, because it has a little to do with applied mathematics; it is impossible to verify the scientificity of many practical quantitative cognitive operations and results in set theory. So, there are far more unscientific contents (more arbitrary quantitative cognizing behaviors) in the quantitative cognitive process of present classical infinite set theory than in present classical mathematical analysis.
Dear Geng Ouyang
You have raised many simultaneous questions about the (finite) and (infinite) concepts. Indeed the latest (infinite) is full of paradoxes in mathematics.
One can observe that :
(*) the part may include the whole,
(*) many infinite sums rearranged to obtain different answers,
(*) Hilbert Hotel paradox,
(*) ambiguity of Cantor sets,
(*) Zeno's paradoxes,
(*) infinity is not real,
All terms: inf - inf, 0xinf, inf/inf, inf^inf, inf^0 1^inf all are undefined!!
So one can stay with such paradoxes years without any clear answer.
And this doesn't mean that we can't use infinity. It is useful to find particular answers for a given mathematical problem. Also, we can construct new definitions that should be consistent with the axioms of the set theory and all other branches of current modern mathematics. All are considered valid based on the added axiom. This is very similar to change the fifth postulate of Euclid's to construct hundreds of non-euclidean geometries; all are consistent and accepted.
So, you can say that the sum of the angles of the triangle is 180 degrees or > 180degree, or < 180 degrees all are correct but in different geometries.
All agree with the initial axioms, but they differ by one axiom.
We can do the same for the set theory.
Dear Mr. Issam Kaddoura , thank you so much.
I agree with you that in some cases, when people look at the things from different angles, some different things can be seen and different ideas and solutions be constructed. But in front of so many old and new suspended infinite related paradox family members having been troubling people for at least 2500 years, we have to consider one question: should those “suspended paradox family members” be removed from human science or they are still necessarily existing in human science and keeping troubling us?
We can see from our mathematical history that so many mathematicians have been trying so hard to remove those “suspended paradox family members” from human science. I have been trying very hard working in this direction and I finally discovered that if we really want to remove them, we have to study the sources of those “suspended paradox family members” and, “removing the fundamental defects — constructing a scientific foundation” is the only way.
It is true that we must inherit much valuable knowledge wealthy from our predecessors in both present classical mathematical analysis and set theory and on the right way of “metabolism”.
Thank you again!
Sincerely yours,
Geng
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