The emergence of new infinite system (new 'infinite' idea, new number system and new limit theory) determines the production of "new mathematical analysis". The mathematical analysis based on the classical infinite system is called "classical mathematical analysis", and the mathematical analysis based on the new infinite system is called "new mathematical analysis (the fourth generation of mathematical analysis)".
There is no infinite related mathematical analysis without the “infinite related number forms”. So, the “quantitative cognizing work on infinite related number forms” is an important and unavoidable task for mathematical analysis. But our studies have proved that in present mathematical analysis, people have been admitting the being of “potential infinite, actual infinite” concepts, unable to deny their qualitative differences and the important roles they play in the foundation of present classical infinite theory system and, unable to deny that the present classical mathematical analysis is basing on present classical infinite theory system. The fact is: on the one hand, present classical mathematical analysis can not avoid the constraining of “potential infinite--actual infinite” concepts and their relating “potential infinite number forms--actual infinite number forms”; on the other hand, no clear definitions for these two concepts of “potential infinite--actual infinite” and their relating “potential infinite number forms--actual infinite number forms” have been constructed since antiquity, thus naturally lead to following two unavoidable fatal defects in present classical mathematical analysis:
（1）it is impossible (unable) to understand theoretically what the important basic concepts of “potential infinite, actual infinite” and their relating “potential infinite number forms--actual infinite number forms” are. So, in many “qualitative cognizing activities on infinite relating mathematical things (infinite relating number forms)” in present classical mathematical analysis, many people actually don’t know or even deny the being of “potential infinite, actual infinite” concepts and their relating “potential infinite number forms--actual infinite number forms”--------the “qualitative cognizing defects on infinite relating mathematical things (infinite relating number forms)”.
（2）it is impossible (unable) to understand operationally what kind of relationship among the important basic concepts of “potential infinite, actual infinite”, their relating “potential infinite number forms--actual infinite number forms” and all the“infinite number forms as well as their quantitative cognizing operations” are. So, in many “quantitative cognizing activities on infinite things (infinite number forms)” in present classical mathematical analysis, many people have been unable to know whether the infinite relating number forms being treated are “potential infinite number forms” or “actual infinite number forms”, no one has been able to avoid the confusing of “potential infinite number forms” and “actual infinite number forms”, no one has been able to know whether or not treating “potential infinite number forms” or “actual infinite number forms” with the same way or different ways. What is more, many people actually don’t know or even deny the being of “potential infinite number forms” and “actual infinite number forms”--------the “quantitative cognizing defects on infinite relating mathematical things (infinite relating number forms)”.
The above two fatal defects have decided since antiquity the absence of “infinite carrier theory” and the confusion of “abstract infinite concept and its concrete infinite mathematical carrier” which have been making us impossible to construct the scientific and systematic theory of “infinite related concrete number form (the mathematical carriers of abstract infinite concept)”, impossible to construct the “potential infinite--actual infinite” relating number forms as well as the “potential infinite--actual infinite” relating number system and its relating treating theory, thus unavoidable forming an insurmountable obstacle in the “quantitative cognizing process on infinite things (infinite number forms)” in present classical mathematical analysis. And, it is impossible to know what those “’both potential infinite number forms and actual infinite number forms’, ‘neither potential infinite number forms nor actual infinite number forms’ mathematical things” are, and can only treat all of them with the “flow line (pipelining)” unified approach （such as the three formal languages in three generations of mathematical analysis: before standard analisis, standard analisis, non-standard analisis）. So, many members of different infinite related paradox families in present “potential infinite--actual infinite” based classical mathematical analysis have been produced one by one naturally (the newly discovered “strict mathematical proven” Harmonic Series Paradox is a typical example), forming a thousands-year old suspended huge black cloud of paradoxes over present classical infinite related science and mathematical analysis.
Our infinite related science history has proved that because of the fatal defects in present classical mathematical analysis basing on “potential infinite-actual infinite”, we are not only unable to solve “infinitesimal related Zeno’s Achilles--Turtle Race Paradox Family” but also unable to solve the Second Mathematical Crisis aroused by “infinitesimal related Berkeley's Paradox Family” within present classical infinite theory system. Three generations of classical mathematical analysis (pre--standard analysis, standard analysis, non-standard analysis) are completely equivalent because of the very same foundation of present classical infinite theory system with “potential infinite-actual infinite”.
Without a sound foundation of scientific infinite theory system, model theory (including the Levenheim-Scolem theorem) can do nothing to solve those suspended infinite related paradoxes produced by the confusion of “potential infinite-actual infinite” in present mathematical analysis and set theory.
An attempt to demystify the notion of infinity:
Andrey Kolmogorov, 1969. Finite creatures in a finite world may be forced to invent infinity.
"... let us imagine an intelligent creature living in a world of finite complexity, [a world] taking only a finite number of physically distinct states and evolutioning in a "discrete time". ... It is possible to explain plausibly how such a creature, because of its structure, being unable to cover all the complexity of the world around it and confronted with systems of growing complexity and consisting of very large numbers of elements, will create during the process of its entirely practically and reasonably oriented activities the concept of an infinite sequence of natural numbers." (Sorry, my own English translation. K. P.)
The original text: A. N. Kolmogorov. Scientific foundations of the school mathematics course. First lecture. Contemporary views on the nature of mathematics, Matematika v Shkole, no. 3 (1969), 12–17 (in Russian). Available online at http://ilib.mccme.ru/djvu/bib-kvant/maths.htm
If the present thousands-year-old model of infinite produces paradoxes, a new one should be created.
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)
The paper “REAL” ANALYSIS Is A DEGENERATE CASE of DISCRETE ANALYSIS
by Doron ZEILBERGER (for a copy see http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf) seems to be relevant to this discussion:
" Continuous analysis and geometry are just degenerate approximations to the discrete world, made necessary by the very limited resources of the human intellect. While discrete analysis is conceptually simpler (and truer) than continuous analysis, technically it is (usually) much more difficult. Granted, real geometry and analysis were necessary simplifications to enable humans to make progress in science and mathematics, but now that the digital Messiah has arrived, we can start to study discrete math in greater depth, and do real, i.e. discrete, analysis."
Thank you dear Dr. Karlis Podnieks,
In fact, not only in analysis, the infinite related number forms in mathematics are very important in set theory too.
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