I do not know the questions you asked, also I do not know, from where did you deduced generations in mathematics? To me, classical mathematical formulations from the very outset to now can have no such categorization as like upgrades in computer software/hardware. In mathematical upgrading, mostly we need to add, explain generalize or approximate classical mathematical formulations for computer to perform computational analysis of the real world via approximate mathematical functions or modelling. Via mathematical functions or approximate models, we performs computations and visualizations etc.

Mathematics is nothing but a scale of universal authenticity regarding finding the behavior of real world systems and objects in the universe, different phenomenons among material objects, etc....That is to perform measurements and experiments on the universal objects and find secrets,. where "distance" is the super-most important function of mathematics ever introduced/investigated.

Dear Mr. Muhammad Ali

I agree with you that it is really not easy to understand this question only by a few lines. But I have uploaded 20 papers about this project to introduce my work, especially 8 papers of ----- .

Studies have proved that present classical mathematical analysis and set theory have been basing on the classical infinite theory system. So, these two mathematical branches have been unable to avoid two fatal fundamental defects: (1) theoretically, confusion on the concepts of “potential infinite and actual infinite”; (2) operationally, confusion on the contents of “potential infinite and actual infinite”. And, these two very defects decide the unavoidable generation of all kinds of infinite relating paradoxes in present mathematical analysis and set theory--------the 2500 years old un-dispersed cloud of “infinite paradoxes syndrome” hanging over the sky of our science. As it turns out, the concepts of “potential infinite” and “actual infinite”, which are un-scientifically defined and mutually contradictory, must be abandoned, and the new infinite theory system basing on ““abstract things—the carriers of abstract things” as well as the systematic theory of “infinite carriers” must be constructed. This is a new way of solving thoroughly the very “infinite paradoxes syndrome” produced by the fatal fundamental defects. The great change of the basic theory in the infinite theory system inevitably leads to the emergence of the forth generation of mathematical analysis and the second generation of set theory.

Thank you.

Sincerely yours,

Geng

If you are asking if it is possible to introduce mathematical analysis without the usual limit concept, more algebraically, I would say you can by now.

The analysis of the infinite beyond Cantor I would say quite tricky or impossible

.

Dear Mr. Juan Weisz,

There are two contradictory cognitive behaviors in present set theory:

On the one hand people deny the necessary relationship between "element” and “set", not knowing at all that the existence of "elements with different properties" leads to the existence of different infinite sets, firmly deny the characteristic differences of elements contained in different infinite sets (the unique existing meaning, unique existing form and unique existing condition as well as unique relationship between and among), and construct a kind of “cardinal number theory” by "double abstraction" which is conflict with the above second nature of the infinite set. Through the "double abstraction", elements contained in many infinite sets are turned into piles of “geometric points" without any differences of “unique existing meaning, unique existing form and unique existing condition as well as unique relationship between and among”, to ensure that they all have the same "cardinal number" making many different infinite sets have the same number of elements. After "double abstraction", elements contained in different infinite sets have lost their original unique properties (including the property of number) and "endless" becomes the only numerical property for all the elements contained in many infinite sets (as we know, all the points on the lines are just piles of "endless abstract things" without any differences of “unique existing meaning, unique existing form, unique numerical property and unique existing condition as well as unique relationship between and among”. All the points on different line segments are with the same "cardinal number" and their quantities are uniformly "infinite"). So, in present classical set theory, after "double abstraction", many subsets and their original sets contain same amount of elements -------- the elements contained in many infinite sets have the same "cardinal number" and "endless" becomes the only numerical property for all the elements contained in many infinite sets.

On the other hand, all in a sudden, people recognized the necessary relationship between "elements and sets", firmly recognized the importance of the essential differences in the manifestation, nature, existing conditions and relationship among the elements in infinite sets, recognized that it is the existence of "elements with unique characteristics" that leads to the existence of different infinite sets; all in a sudden, people denied the "double abstraction theory”, suddenly decided not to apply "double abstraction theory” in the cognitive process for elements contained in the infinite set, so as to ensure the applying of T = {x|x📷x}theory which has nothing to do with "double abstracted” to find some elements still with their unique original features (not being "double abstracted”) and to complete some proofs that some infinite sets contain more infinite elements than other sets (for example, the infinite elements contained in Real Number Set are more than the infinite elements contained in Natural Number Set, the infinite elements contained in any infinite set are less than the infinite elements contained in its power set -------- Infinite Real Number Set is more infinite than Infinite Natural Set Number Set, and any infinite set is less infinite than its power set,... As the matter of fact, in present number system (number spectrum) of our mathematics, there is no such number form as "cardinal number" without specific quantitative meaning. Studies have shown that, the unavoidable conceptual confusion of "potential infinite, actual infinite" in present classical infinite theory system determines Cantor's theory and operations of “cardinal number” and "double abstraction" are not self-justification at all and lack of scientific and systematic, which inevitably results in the impossibility of scientific, effective and systematic qualitative and quantitative studies to elements contained in infinite sets.

The fatal fundamental defects in present classical infinite theory system are the source of the second and the third mathematical crisis: more than 2500 years, no one can get rid of a kind of disease in the infinite related fields of mathematics --------- a diseases produced by the confusion of "potential infinite and actual infinite" concepts in set theory diagnosed clearly by Poincare, Frege, and Weyl more than 100 years ago. Studies have proved that this is the common "disease" existing in many infinite related mathematical disciplines with the foundation of present classical infinite theory system: the various "number and non--number mathematical things” -------- “variables of not only potential infinite but also actual infinite (the ‘ghost’ disappearing and reappearing at any time?)" for all the family members of Zeno's Paradox and Berkeley's Paradox in mathematical analysis [1-6]; the mathematical things with the property of "elements belonging and not belonging to a set ---------- T = {x|x📷x} (variable elements of not only potential infinite but also actual infinite: the ‘ghost’ disappearing and reappearing at any time?) " for all the family members of Russell paradox in set theory；…

This is why we are so sure to say that the Third Mathematical Crisis in present scientific theory system is unsolvable and the Third Mathematical Crisis is another manifestation of the Second Mathematical Crisis in set theory. They are "twins".

Thank you.

Sincerely yours,

Geng

Im not surprised by the confusion, the infinite sets have so many properties not ascribable to finite sets. A piece of straight line has infinite points, but a shorter and a longer piece have the same number. You have the same problem with numerable infinite sets.

In these cases the infinity is real, not potential.

You cannot discuss from the potential, because you would have to start from finite sets, completly different from infinite ones.

At least for all finite sets, the elements obviously define the set

would be what I think. Even mentioning an infinite set, you have also to define it in some way.

Dear Mr. Juan Weisz,

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?

Our studies have proved that the quantitative cognizing theories and operations (including the theory and operations of one to one correspondence and limit theory) for those infinite related mathematical things in The First Generation of Infinite Set Theory are lack of scientific foundations: It is impossible to know at all what the relationship among all the quantitative cognizing behaviors in infinite set theory and the concepts of "potential infinite” and “actual infinite" is and how to carry out scientific and effective operations specifically to different kinds of infinite related mathematical things in infinite sets. 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. Therefore, 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.

Thank you.

Sincerely yours,

Geng

Again and again you start a question basing on my research. When I give you an answer you neither argue against my assertions nor you accept my research. This is not how scientific discussion works. It´s more near or like plagiating the research of others — mine! Read it and discuss or leave it. You do not have the authorship of these questions, so you have to name the sources (me / the address), because you know. Since 2016 the papers on Cantor are at my homepage (www.mathe-neu.de) avaiable for free. Some days ago also at RG (https://www.researchgate.net/project/Logic-of-Formalism-How-physics-raises-math-to-a-natural-science). See articles 03 / 04 / 05. Regards, Peter

Dear Mr. Peter Kepp,

I began the infinite related project 45 years ago, and my first paper on this project was published in 1995 and my first paper on set theory was published in 1998.

Would you please check my published papers on the net of CNKI and those I uploaded on RG?

I am very sorry to say we two have totally different working thought and different results.

Thank you.

Sincerely yours,

Geng

Dear Mr. Geng Ouyang,

would you please be so kind and give me a complete address for a paper (written in English) you are the author of and which is on Cantor (actual infinite / potential infinite). I will read. We came together often the last time on that topic.

If two are working at the same topic, different results should get discussed. In my opinion I proved that there is no need for any extended property to `infinite´.

Regards, Peter

The advantage of the potentially infinite, to achieve the infinite by some limit of the finite, is that there would not be this abrupt break with properties of finite sets.

So finally it seems to me there can be two radically different approaches to infinite sets.

This is consistent with the view of many analysts, who say infinity as a number does not exist, potential infinity is just as large as needed in each step. This is the view of intuitionists also. Somehow this sets a limit to methods of induction.

Dear Mr. Peter Kepp,

I am sorry that I have not translated all of my 58 published papers because it is really a hard work. Just read my following published papers being uploaded onto RG, you will understand most of your problems.

1, On the Quantitative Cognitions to “Infinite Things” (VIII)------------ Three Exactly Same Fundamental Characteristics in the Old and New Generations of Infinite Set Theory and Mathematical Analysis

2, “A Newly Constructed Infinite System”--------Forty-year Work in the Infinite Related Foundation of

Mathematics

3, A Revolution in the “Infinite” Related Foundation of Mathematics-------“New infinite conception--new

infinite number system--new limit theory” and Solving “Infinite Paradox Syndrome”

4 Four Errors in Cantor’s Proofs on the Uncountability of Real Number Set and the Defects in the

Infinitude Related Foundation of Mathematics

Thank you.

Sincerely yours,

Geng

Dear Mr. Juan Weisz ,

Thank you for your insightful idea.

So many old and new paradox family members have been being right there in both present classical infinite related mathematical analysis and set theory. So, it is believed that whether or not to solve them and prevent them is a touchstone.

The new concepts of "abstract things and the carriers of abstract things" and the systematic theory of “he carriers of abstract things” being introduced can get rid of most of paradoxical factors in both present classical mathematical analysis and set theory.

Sincerely yours,

Geng

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Dear Mr. Geng Ouyang,

I have read yout paper `Four errors ...´. I trust you for having published it before you could read my papers. So I was wrong with any reproach. Please excuse me.

You did a lot of work. Some is rght, some is wrong but it is definitely not complete. So please read my papers too for to come to a fruitful discussion.

Right: Construction of Cantor seems to produce only decimal-numbers (precise: only numbers of a single measure — base 10: n10).

Right: `Vicious Circle´. [but that could be explained better]

Wrong: The one-to-one ... between the naturals and the sequences Cantor constructed is possible!

??? What about the abstract signs Cantor used?

??? Where is your comment on the relationship between /mu (greek letter) and /nu (greek letter) — Cantor had used as the two indexes?

??? Why didn´t you see that the numbers (base 10) — Cantor produces at his sequences (by naming no other measure) — are the complete set of his first diagonal argument? [so countable; limitation limits the effect]

Wish to hear from you, best, Peter

Dear Mr. Peter Kepp,

Just go ahead according to your own working thought.

Sincerely yours,

Geng

There is nothing for `to go ahead´ if a complete proof is done except a discussion.

Are you not able to read / study my papers?

https://www.researchgate.net/publication/338390480_Logic_of_Formalism_03_-_Cantor?_sg=Pypw1NOLcZRZj5PYP9ogic6JuQlJOYX0-qHmIXVlUzcxZLcPQqoBjKZOlmxhPXEz8Oh-G30zTC2TM2FFez_kEWh_XhAmmuGb8jmfCfq4.71A31TlrsTmpV_-AtirzxkR276J3GhJNxVnPLtw_UGyO2nf6ZZQdZgghYWkyPTCEY6pyOJdU5WHs-HStjjmaiw

https://www.researchgate.net/publication/338390666_Logic_of_Formalism_04_-_Proof?_sg=Pypw1NOLcZRZj5PYP9ogic6JuQlJOYX0-qHmIXVlUzcxZLcPQqoBjKZOlmxhPXEz8Oh-G30zTC2TM2FFez_kEWh_XhAmmuGb8jmfCfq4.71A31TlrsTmpV_-AtirzxkR276J3GhJNxVnPLtw_UGyO2nf6ZZQdZgghYWkyPTCEY6pyOJdU5WHs-HStjjmaiw

https://www.researchgate.net/publication/338390579_Logic_of_Formalism_05_-_Cantor_3?_sg=Pypw1NOLcZRZj5PYP9ogic6JuQlJOYX0-qHmIXVlUzcxZLcPQqoBjKZOlmxhPXEz8Oh-G30zTC2TM2FFez_kEWh_XhAmmuGb8jmfCfq4.71A31TlrsTmpV_-AtirzxkR276J3GhJNxVnPLtw_UGyO2nf6ZZQdZgghYWkyPTCEY6pyOJdU5WHs-HStjjmaiw

What makes you sure beeing right? You only (as others) discovered the vitious circle in the rule of Cantor producing the diagonal line as different to the source-lines it gets developed from (the /mu power /nu problem). But you are not complete (in relationship to whole the problem) at your conclusion.

`Just go ahead ...´ seems very less interested in the topic for getting more insight or interest for any discussion with well informed others.

Regards, Peter

Dear Mr. Peter Kepp,

I am sorry to say it seems you are forcing me to do something. To say the truth, I don’t appreciate such a way in doing scientific discussions.

Well, don’t be worry that much about the “mistakes” in Cantor’s proofs on the uncountability of real number set; anyhow, it is only a tip of the iceberg in the huge infinite related science project.

Sincerely yours,

Geng

G. O.: ”... forcing me to do something. ”

Yes, first to study, then to make research. What about trying to understand the first diagonal argument of Cantor?