Dear Geng,

I can't quite understand your language. The survival of Mathematical theories does not depend on their ages, but their consistency. This is why I cannot consider the terms "old" and "new" as logical axioms of validity.

Dear Geng,

I can't quite understand your language. The survival of Mathematical theories does not depend on their ages, but their consistency. This is why I cannot consider the terms "old" and "new" as logical axioms of validity.

Although Dr.Juan-Esteban Palomar Tarancon is formally right, the problem is still very interesting. The "seditious" question of the applicability of "naive" arithmetic operations to cardinals and ordinals seems to be strongly related to the questions of model theory, in particular the Levenheim-Skolem theorem, and the concept of a fractal of numerical systems (see my project under this title).

Thank you very much dear Dr. Juan-Esteban Palomar Tarancon and Dr. Alexey Orlovsky for your constructive opinions.

I have corrected some mistakes in my English sentences above. I beg your pardon for my poor English.

Yours,

Geng Ouyang

Are there more than two major differences between the classical and new infinite set theories (The Second Generation Of infinite Set Theory)?

It is well known that the founding of present classical infinite set theory is a brilliant achievement in our science history. Cantor leaded us human into a new field of “quantitative cognitions to infinite related mathematical things”. “’Infinitesimal, infinite greats and infinite many’ related mathematical things” should be qualitative-quantitative studied and cognized ------- infinite set theory is needed and there must be infinite set theory in human science.

The emergence of new infinite system determines the production of "new infinite set theory"--------The infinite set theory based on the classical infinite system is called "classical infinite set theory " while the infinite set theory based on the new infinite system is called "new infinite set theory (the second generation of infinite set theory)".

Different basic theories decide the some differences between new and classical infinite set theories.

1, Different infinite ideas

In classical infinite set theory, “infinite” is composed by both contents of “potential infinite, actual infinite”. So, any infinite related “mathematical things” in present classical set theory basing on classical infinite idea can not run away from the fetters of “potential infinite, actual infinite” [13-22]. Although it is generally accepted that “actual infinite” is a kind of “tangible infinite”, no one in the world so far knows exactly how on earth “actual infinite” be “tangible” theoretically and operationally, and no scientific and clear definition has ever been given for “potential infinite-- actual infinite” since they came to human science and mathematics. The inevitable confusion of “potential infinite, actual infinite” and the absence of the whole “actual infinite theory” in the “potential infinite, actual infinite” based present classical set theory inevitably result in following fatal defect: it is impossible to know clearly what on earth the exact “infinite mathematical things” being quantitative cognized in set theory are and how to be treated (such as “are they potential infinite mathematical things or actual infinite mathematical things?”, “whether “potential infinite mathematical things or actual infinite mathematical things” should be treated differently?”, “if they should be treated differently, how to operate?”, “why?”, …). So, there are no scientific basis for many quantitative cognizing activities to “infinite mathematical things” in present classical set theory------- just do as one wishes to produce many inevitable errors and paradoxes.

In new infinite set theory, “infinite” is composed by both contents of "abstract infinite law and the carriers of abstract infinite law” (called “infinite” and “infinite carrier” for short following) which can be scientifically and clearly defined without any confusion [7-22]. The “infinite carrier” in new infinite idea is a kind of “visible and tangible science thing” and it is possible for us to construct a complete and systematic “theory of infinite mathematical carriers”------the infinite related new number forms (number carriers), new number spectrum, new set forms (set carriers), new set spectrum as well as the treating theories for those “infinite mathematical carriers”. This new “theory of infinite mathematical carriers” enable us to know theoretically and operationally what on earth many “infinite mathematical things (infinite carriers)” being cognized are, how they are “visible and tangible” and how to carry on the scientific operations of quantitative cognizing to them. So, it is impossible to have any theoretical and operational confusions of “potential infinite-- actual infinite” in new infinite theory system and, the “‘infinite paradox’ symptom complex” produced by the “confusions of potential infinite-- actual infinite” in present classical set theory is to be naturally eliminated.

2 Different quantitative cognizing operations and results to "infinite related mathematical things"

In classical infinite set theory: nothing is able to run away from the constraining of “potential infinite--actual infinite” and all of its “mathematical things" can only be existing within following three "potential infinite--actual infinite" related contents: (1) "potential infinite mathematical things (such as potential infinite numbers, potential infinite sets, potential infinite elements of infinite set)" ; or (2) " actual infinite mathematical things (such as actual infinite numbers, actual infinite sets, actual infinite elements of infinite set)"; or (3) “the mixture mathematical things of potential infinite and actual infinite" which can be either "potential infinite” or “actual infinite" as one wishes at any time. All these “infinite related mathematical things" in present classical set theory have nothing to do with present classical number system and set system (there are not any “potential infinite--actual infinite” related number forms and set forms in present classical number system and set system), nothing to do with the specific quantitative properties of "infinite mathematical things" (What quantitative properties do potential infinite numbers, potential infinite sets, potential infinite elements of infinite set have? What quantitative properties do actual infinite numbers, actual infinite set, actual infinite elements of infinite set have? Can these infinite mathematical things be numbers or sets? ...). Such fundamental defects have been making us unable to understand scientifically and clearly the quantitative properties, meanings and forms of infinite mathematical things in present infinite related classical mathematics system.

So, theoretically or operationally, it is impossible for us to establish the scientific and systematic quantitative cognizing and operating theories on “infinite related mathematical things (such as quantitative cognizing and comparing theories for those "X----> 0 infinite related mathematical things and Y ---->📷infinite related mathematical things", the operating theories of one-to-one corresponding theory and limit theory, ...)” in present classical “potential infinite--actual infinite” based set theory. Such fatal fundamental defects naturally result in following two unscientific contents in present classical set theory:

（1）arbitrary infinite theory: only accepting infinities but denying infinitesimals; “infinite sets, elements of infinite sets, numbers of elements in infinite sets” can be taken as "potential infinite” or “actual infinite" or the mixture of both; ...);

（2）arbitrary operations and formal languages: any “infinite series” can be cut into pieces to produce super infinity, super super infinity, super super super infinity,... with different grades; never knowing how and when to use the formal languages and operations of “let be 0, take the limit, take the standard number” for those “X----> 0 infinitesimal elements” in [0,1] Real Number Set; an exactly same pipeline operation and formal language can be used to produce totally different products------- the “one-to-one corresponding operation and formal language” can not only be used for the result of “Rational Number Set is equal to Natural Number Set” but also for the result of “Rational Number Set is bigger than Natural Number Set”, “the pipeline operation and formal language of looking for something belongs to an infinite set but is impossible to be found inside this infinite set (the confusing mechanism of ‘potential infinite--actual infinite’)” can not only be taken as “fatal fundamental defects” in present classical set theory resulting in the production of Russell’s Paradox, but also as “important fundamental theory” in present classical set theory resulting in the production of modern “Power Set Theorem” and “Uncountability of Real Number Set Theorem” , ...

In new infinite set theory: the “new infinite carriers’ theory” closely relating to “new number forms, new number system (new number spectrum), new set forms, new set system (new set spectrum)” is an important basic theory for us in our quantitative cognizing on “infinite related mathematical things”. All the “infinite related mathematical things” being cognized are “mathematical carriers of abstract infinite law” with tangible quantitative properties and meaning, expressing the existence of abstract infinite law-------they are “mathematical things” with unequivocal quantitative properties and manifestation modes clearly defined by new number spectrum and set spectrum (such as inter-numbers and inter-sets). So, different from the inevitable confusion situations of “potential infinite” and “actual infinite” in classical infinite set theory, it is impossible in new infinite set theory to have the confusion situations of “abstract infinite law” and “carriers of abstract infinite law”. But, the most important thing is, it never allows taking “law” as “carriers” for quantitative cognition like the classical infinite set does. Being one of the “science carrier theories”, the newly developed “theory of infinite related mathematical carriers” as well as its relating “carriers analyzing and treating theories (such as ‘new limit theory’ and ‘new one-to-one corresponding theory’ relating to ‘infinite related mathematical carriers’)” fills in a fundamental blank for the “quantitative cognizing theory for infinite related mathematical things” in present classical mathematical analysis and set theory and open up a new quantitative cognizing field of “infinite related mathematical things”. The new basic theory ensures the Second Generation of Set Theory be free from any unscientific influences of "potential infinite--actual infinite" and eradicates above unscientific “paradoxes producing factors” as well as their production of “infinite paradox symptom complex”. Before practical quantitative cognizing operations in new set theory, those cognized “mathematical things” must be truly "mathematical analyzed" according to the newly developed “theory of infinite related mathematical carriers” to know what quantitative natures, meanings and manifestations they have so as to perform the targeted quantitative operations to them.

So, such targeted operating mechanism. method, process and result decided by the newly developed “theory of infinite related mathematical carriers” as well as its relating “carriers analyzing and treating theories” in new infinite set theory are surely nonequivalent in many situations to those operating mechanism. method, process and result decided by “potential infinite--actual infinite” in present classical set theory-------it is impossible for the Second Generation of Set Theory to have (to produce, nourish and protect) those suspended paradox families relating to the quantitative cognitions of "infinitesimals and infinities" as the first Generation of Set Theory has.

Except those unscientific contents caused by the confusion of “potential infinite--actual infinite”, the Second Generation of Set Theory retains most of the things in the first Generation of Set Theory.

The 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, ZFC, 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.

Happy Chinese Spring Festival!

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)