How can we define “infinite” and “finite” if we agree with the idea and the operations in Cantor's proofs that many infinite sets in mathematics actually can be proved (turned into) finite set-------the elements in real number set are never-be-finished, endless, limitless and they are really infinite while those in natural number set are sure-be-finished, ended, limited and they are actually finite?
This conundrum has disclosed two infinite related fundamental defects in present infinite set theory and mathematical analysis:
（1）the theoretical and practical confusions of “finite--infinite” caused by the theoratical and practical confusions of “potential infinite, actual infinite”;
（2）the absence of systematic “scientific carrier theory” and the absence of systematic “infinite related number spectrum and set spectrum”.
Because of the absence of systematic “scientific carrier theory”, in present infinite theory system (The First Generation of Infinite Theory System) basing on “potential infinite, actual infinite”, people have had to use the unscientific “potential infinite, actual infinite” theory for the quantitative cognitions of those “finite--infinite” related mathematical things since antiquity. And, because all the “infinite related concepts and contents” for quantitative cognizing in our mathematics have been being unscientifically defined as “non—infinite infinite, finite infinite, infinite finite, both finite and infinite, number of non-number, ...” in present infinite theory system, no one is able to know clearly when and where they should be “finite or infinite, number or non-number, ...” and why. This situation is not only unable us to cognize scientifically two concepts of “finite, infinite” and the relationship between them, but also unable us to cognize scientifically three contents of “finite mathematical things, potential infinite mathematical things, actual infinite mathematical things” and the relationship among them in present infinite related mathematical fields------very free and arbitrary when dealing with any “infinite related mathematical things”. So, many suspended infinite related paradox families have been produced inevitably. It is because of such fatal fundamental defects that results in the frequently performings of the mathematical version of Andersen's THE EMPEROR'S NEW CLOTHES Fairy Tale in all infinite related mathematical fields ever since “potential infinite, actual infinite” came into our science and mathematics, the only thing most people are able to do is just echoing in earnest to what the authorities said.
In new infinite theory system basing on the new concepts of "abstract infinite law and carriers of abstract infinite law " (The Second Generation of Infinite Theory System), the concepts of “finite, infinite, finite scientific carrier, infinite scientific carrier” have been clearlly and scientifically defined. “Infinite” is a kind of “invisible and intangible abstract scientific law”, it is a scientific (mathematical) content for qualitative but not quantitative studies and cognizings; while the “abstract scientific infinite law carriers" is a kind of “visible and tangible scientific concrete manifestation (entity) of abstract scientific law”, it is a scientific (mathematical) content specially fot quantitative studies and cognizings. In our science, we can only conduct the quantitative studying and cognizing on the “abstract scientific law carriers" but not on the “abstract scientific law". Basing on new infinite theory system, now we are perfectly justifiable to construct a systematic and scientific “theory of infinite related mathematical carriers" and able to conduct quantitative studying and cognizing on all kinds of “infinite related mathematical carriers" scientifically (such as their computations as well as the comparetions among them). The quantitative related systematic and scientific “theory of carriers" decides that we must have quantitative studies and cognizings on “different infinite law carriers, big (small) infinite law carriers, bigger (smaller) infinite law carriers, more bigger (more smaller) infinite law carriers, ..." but not on “different infinite laws, big (small) infinite laws, bigger (smaller) infinite laws, more bigger (more smaller) infinite laws, ..." or even arbitrarily on the “finite—infinite mixing up things” as what have been done in presentmathematical analysis and set theory (such as taking them as finite things first then as infinite things later or taking them as infinite things first then as finite things later or the elements belonging to an infinite set but being impossible to exist in this very infinite set, ...). So, in the Second Generation of Set Theory, it is impossible at all to have such contents as “the infinite of more infinite, more more infinite, more more more infinite, ... super infinite, super super infinite, super super super infinite, super super super super infinite, ...”; All the mathematical contents of “X--->0, Y--->📷, ...” for quantitative cognizing in our new infinite related mathematical analysis and set theory are the “scientific carriers" with clear and scientific quantitative natures and definitions in new systematic and scientific “infinite related number spectrum (number system) and set spectrum (set system)”. The replacement of “potential infinite, actual infinite” concepts by new "abstract infinite law and carriers of abstract infinite law" concepts in present infinite theory system and the development of systematic “theory of scientific carriers” inevitably result in great theoretical and operational reformations (a revolution) in the quantitative--quantitative studies and cognizings in our infinite related science areas and, eliminate scientifically and thoroughly the suspended conundrums of “’potential infinite, actual infinite’ confusing" and “’finit--infinite’ confusing" since antiquity.
Thank you for your idea dear Mr. Aleksandar Jovanovic.
I wish to disclose the theoretical and practical confusions of “finite, infinite, potential infinite, actual infinite” in present infinite related science and mathematics. This is the very fundamental defect for the generating of those paradox families (infinite paradoxes symptom complex) in present mathematical analysis and set theory.
Gang, I think that problem is ( partially) with the use of a language. To get clear understanding what a particular language expression mean you need a higher ( the second) level f language ,or meta language to describe the first level language . This in turn will call for a meta meta language and so on. Those are difficult problems and I have not enough expertise in that field to comment any further.
The idea of meta language is very good, at the same time we can speak of meta meta consciousness and who wil who meet at this meta meta meta point.
Thank you very much for your ideas and suggestions dear Mr. J. G. von Brzeski and Ms. Rita De Vuyst,
First of all, I beg your pardon for my poor English.
The paradox families (infinite paradoxes symptom complex) in present mathematical analysis and set theory have been being right there since Zeno’s time. I am trying another way to solve the fundamental defects disclosed by them.
Five of my published papers have been up loaded onto RG to answer such questions:
1，On the Quantitative Cognitions to “Infinite Things” (I)
2，On the Quantitative Cognitions to “Infinite Things” (II)
3，On the Quantitative Cognitions to “Infinite Things” (III)
4 On the Quantitative Cognitions to “Infinite Things” (IV)
5 On the Quantitative Cognitions to “Infinite Things” (V)