The 'size' of a set is estimated by means of a bijection between that and some previously selected set. So, he can not look at the 'size' of a set without compromising with another set. Your use of the syntagm 'half of the Natural Number Set (2,4,6, ..., 2n, ...)' is incorrect. Set (2,4,6, ..., 2n, ...) is a subset of the set {1,2,3, .., n, ...} and nothing else In the Set theory, we know, injection / surjection / bijection is a tool for comparing sets.

Thank you for the question and the authors of this ... Greetings

Dear Dr. Daniel Romano, thank you.

I now use “a subset of the Natural Number Set ” instead of syntagm 'half of the Natural Number Set’.

Yous,

Geng Ouyang

Thank you Dear Dr. Aleksandar Jovanovic.

Yous,

Geng Ouyang

Thank you Dear Dr. Jaykov Foukzon .

Yous,

Geng Ouyang

The correct terminology is "integers" not "natural numbers", there is nothing "natural" about the integers. The integers is a proper subset of the rational numbers which in turn is a proper subset of the real numbers. None of these sets is finite (finite means that you can count the members of the set and there is a last integer used in the counting). The integers is "countable" but so is the set of rational numbers (a rational number is the quotient of two integers), However the set of real numbers is not "countable", i.e. there is no 1:1 correspondence between the set of the real numbers and the set of integers.

You can not think of infinite sets in the same way as you can for finite sets, using the word "size" is completely mis-leading and inappropriate. You will not understand if you try to use that term with respect to infinite sets.

Instead you have to talk about the "cardinality" or "ordinality" (the latter pertains to "ordered sets". These are not intuitive terms, they have very specific definitions. Two sets have the same cardinality if there is a 1:1 correspondence between them but the correspondence may not preserve the order.

Square root of 2 is an example of an irrational number and is also a real number but there are real numbers that are neither rational nor irrational, for example "pi".

"infinity" is not a "number", it is a concept or idea. You can't treat it like a number (as in arithmetic).

You need to consult a book on Set Theory or on Real Analysis.

Thank you for your knowledge from books Dear Dr. Jaykov Foukzon.

How do you think of following two questions:

1, Are the concepts of “potential infinite--actual infinite” important in present classical infinite related science system? If yes, what roles they play; if not, why they have been existing in our science?

2, Is it possible to produce infinite items (numbers) each bigger than 1/2 or 100 or 1000000000000000 or 1000000000000000000000000000000 or… from the Harmonic Series 1, 1/2, 1/3, ... by the operation of "bracketing” with limit theory and turn the Un--->0 Harmonic Series into a “Vn ---> any positive constants” infinite series with infinite items each bigger than any positive constants (such as 100000000000000000000000000000)?

Yous,

Geng Ouyang

To Geng Ouyang

Unfortunately none of your comments, e,g, "infinite items", "classical infinite related science system", "potential infinite", "actual infinite" make any sense. You are misinterpreting the ideas in the paper by Sergyev.

Yes the set of even integers is a proper subset of the set of integers but the cardinality is the same for both sets. There is a 1:1 correspondence between these two sets, Neither set has more nor fewer members than the other.

The explanation given by D. Romano is all wrong

Thank you for your intresting idears, Dr. Donald Myers .

Yous,

Geng Ouyang

There are at least three reasonable non-equivalent definitions of cardinals. Please, remember that it is impossible to prove that the set of all even non-negative integers is of the same cardinality as the set of all non-negative integers without suitable axioms of a set theory nor without making it more precise what the set of all non-negative integers is, nor without saying in what sense the word "cardinal" is to be used. Axioms are only some assumptions. Theories of proofs can be very complicated. I recommend the book "The Foundations of Mathematics" by K. Kunen. I like this book very much.

Regards, Eliza Wajch

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.

At the risk of prolonging a nonsensical discussion, Geng Ouyang's comments are complete nonsense;

I agree with Prof.Daniel .

Are there more than two similarities between new infinite set theory (The Second Generation of Infinite Set Theory) and classical infinite set theory (The First Generation of Infinite Set Theory)?

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 (the First Generation Of 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)".

The same mission and same cognizing contents decide the two similarities between the two set theories.

1 The same qualitative-quantitative cognizing motivation and idea

It is a must to carry on the qualitative-quantitative cognizing activities on “infinite related mathematical things (such as elements in infinite sets and the quantity of elements in infinite sets)” by both new and classical set theories; especially in many practical quantitative cognizing operations, most “mathematical contents” in infinite sets are treated as “mathematical things with visibal and tangible quantitative nature and meaning” by both new and classical set theories.

2 the same quantitative cognizing tools

Both new and classical set theories use one-to-one coresponding theory and limit theory to carry on quantitative cognitions to those “infinite related mathematical things” in infinite sets.

It is these two similarities between new and classical set theories that decides many invaluableners intellectual wealth accumulated since antiquity in classical set theory are reserved in new set theory.

At first glance I don't understand your question. I assume your question is:

There is a subset of Natural Numbers bigger or smaller than other subset of Natural Numbers, including Natural Numbers itself?

The short answer is no. All non-finite subsets of Natural Numbers have the same size, the size of set Natural Numbers. This propriety is paradoxical an was used as a definition of an infinite set.

Def: A set is "infinite" if can put itself in correspondence with a subset of itself.

Then Natural Numbers is a "infinite set" because it can put in correspondence with a subset of itself like the even or odd numbers with all Natural Numbers.

Now the term "Natural Number" is correct because in set theory is the first minimal infinite set that we can construct using set theory. An then the things worsening.

The next sets of numbers have the same size than the natural numbers. The following numbers have the same size: Natural Numbers, Integers, Rationals and Algebraic. The set of Real Numbers is bigger than Natural Numbers.

The symbol to denote the size of natural Numbers is ℵ0 and is called the first transfinite number. This number is the cardinality of Natural Numbers, how many. The "second" first transfinite is ω. This is how the numbers are arranged. In finite sets the ordinal and cardinal number is the same number, but in transfinite have different meaning.

The next transfinite numbers are ℵ1 and ω1. But nobody knows if ℵ1 and ω1 are the same that the set of Real Numbers and this is the continuum hypothesis. If the continuum hypothesis are true ℵ1 and ω1 are the transfinite numbers of Real. If the continuum hypothesis are false ℵ1 and ω1 are different number, and smaller that the Real.

About the "first and second set theory" I assume you are talking about the Cantor theory and the axiomatization of Zermelo, Frenkel and others, known as ZF or ZFC if choice axiom is included.

The necessity of new set theories was aroused because Cantor set theory have a paradoxical results, and therefore some mathematicians and philosophers develop a new set theory.

To know more of this see:

https://plato.stanford.edu/entries/set-theory/

https://plato.stanford.edu/entries/settheory-early/

https://plato.stanford.edu/entries/settheory-alternative/

Thank you dear Mr. Mauricio Algalan Meneses,

The fundamental defects of “potential infinite and actual infinite” confusions in present classical infinite set theory have been making us humans unable to study and cognize scientifically the foundation of “one-to-one correspondence theory” (the “one-to-one correspondence theory needs its own foundation” even have never been considered about). And, because of the absence of this very foundation, it is very difficult for people to really understand scientifically what kind of mathematical tool “one-to-one correspondence theory” is and how to operate with this mathematical tool in practical quantitative cognitions to elements in infinite sets. So, following five questions have been produced and troubling people long:

（1）Are the elements in infinite sets “potential infinite things” or “actual infinite things”？

（2）Are there different “one-to-one correspondence theories and operations” to “potential infinite elements” or “actual infinite elements” in infinite set theory？

（3）How do we practically carry on “one-to-one correspondence” operations between two sets-------do we have “‘one single element’ to ‘one single element’ correspondence” or “‘one single element’ to ‘many elements’ correspondence” or “‘many elements’ to ‘many elements’ correspondence”？can we arbitrarily alter the elements’ “special nature, special existing condition, special manifestation and special relationship among each other” in infinite sets during the “one-to-one correspondence operations” for quantitative cognitions (such as alter all the elements in Natural Number Set first [1x2, 2x2, 3x2, 4x2, …,nx2, …] = [2,4,6,8, …,e,…] (not the correspondence between N and E but E andＥ), then prove it has same quantity of elements in Even Number Set)?

（4）What kinds of the elements in two different infinite sets are corresponded-------- do we have “‘one single original element’ to ‘one single original element’ correspondence” or “‘actual infinite elements’ to ‘potential infinite elements’ correspondence” or “mixture correspondence of ‘actual infinite elements’ and ‘potential infinite elements’”？

（5） What on earth is the foundation of “one-to-one correspondence theory”？

The fundamental defects in present classical infinite set theory have made us unable at all to answer clearly and scientifically above five questions. So, when carrying on practical quantitative cognitions to elements in different infinite sets with “one-to-one correspondence theory”, one can do very freely and arbitrarily--------lacking of scientific basis. For example: it is because of acknowledging the differences of elements’ “special nature, special existing condition, special manifestation and special relationship among each other” between Real Number Set (R) and Natural Number Set (N), one can prove that the Real Number Set (R) has more elements than N (the Power Set Theorem is proved in the same way). But, as what has been discussed in above 2.1 .1, we are able to prove with exactly the same way “the mother set has more elements than its sub-set”, “Rational Number Set has more elements than Natural Number Set”, “Natural Number Set has more elements than odd number set” ,...; we can even apply the widely acknowledged method of altering elements’ “special nature, special existing condition, special manifestation and special relationship among each other” to prove “Natural Number Set has more elements than Natural Number Set”, “odd number set has more elements than even number set”, “even number set has more elements than odd number set”, ....

Basing on the new infinite theory system with the “infinite mathematical carriers theory”, the Second Generation of Set Theory provides us with the scientific foundation of “one-to-one correspondence theory” and enable us answer above five questions clearly and scientifically:

（1）the elements in infinite sets are “infinite related mathematical carriers” with explicit quantitative nature and definition, indicating the existing of “abstract infinite law” and nothing to do at all with “potential infinite--actual infinite”. This decides one of the major differences between the first and the second generation of set theories-------the elements in different infinite sets have their own “special nature, special existing condition, special manifestation and special relationship among each other. So, it is really possible that different infinite sets have different quantity of elements and people can take them really as “visible and tangible infinite related mathematical things (such as the new numbers in new number spectrum)” for the quantitative cognitions

（2）the elements in infinite sets have nothing to do at all with “potential infinite elements” and “actual infinite elements”, there is only one identity for them-------“infinite related mathematical carriers” with explicit quantitative nature and definition; So, there is only one “one-to-one correspondence theory and operation” for them.

（3）it is explicitly stipulated that only “‘one single original element’ to ‘one single original element’ correspondence” operation is scientific (allowed) when comparing two sets for the quantitative cognitions and, during this process, any operations of arbitrarily altering the elements’ “special nature, special existing condition, special manifestation and special relationship among each other” are unscientific (not allowed).

（4）in the Second Generation of Set Theory, because of nothing to do at all with “potential infinite--actual infinite”, it is impossible to have any troubles produced by the confusion of “potential infinite --actual infinite”.

（5）the new infinite theory system (especially its theory of “infinite related mathematical carriers”) is the foundation of “one-to-one correspondence theory”.

Yours,

Geng Ouyang

I believe you have a more deeper question at the bottom.

¿How can use the infinite sets and how are they?

Today most of mathematicians do not distinguish between potential and actual infinite, because Cantor and then ZF or ZFC theories legitimize the use of all kind of infinite, or properly transfinite numbers. This follows the philosophy of mathematics used by Hilbert.

There is a small group of mathematicians who distinguish between potential and actual infinite. They use the intuitionist philosophy of mathematics, investigated by Poincaré, Brower and others. They believe there is only valid transfinite number ℵ0 and ω, potential infinite. The next transfinite to them is the "continuum" or actual infinite.

Till today are 3 mayor schools of philosophy of mathematics:

To know more who they conceive the infinite you can consult the Stanford Encyclopedia of Philosophy. A resume is here:

https://plato.stanford.edu/entries/philosophy-mathematics/

Note: The article says are four schools but the last 'predicativism' do not have many followers.

And an excellent but specialized book is:

The Oxford handbook of philosophy of mathematics and logic

Authors: Stewart Shapiro Publisher: Oxford University Press Description: Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge-gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines.

A preprint version is here:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.464.6584&rep=rep1&type=pdf

The earlier recommended bibliography to you looks good. Consult the references if you want deepen in this topic.

Thank you dear Mr. Mauricio Algalan Meneses,

Following infinite related questions have never been answered clearly and scientifically since the concept of “infinite” came into human science:

Are the concepts of “potential infinite--actual infinite” important in present classical infinite related science system? If yes, what roles they play; if not, why they have been existing in our science ever since? How can we cognize the relationship between “infinite related mathematical things” and “potential infinite--actual infinite”?

So many people have been trying to run away since antiquity from the concepts of “potential infinite--actual infinite” in present classical infinite theory system which are impossible to be defined scientifically and really “troubles making”, but our thousands—year infinite related science history has proved this is impossible at all. Thus, following suspended contradictions have been inevitably existing in present infinite related set theory and mathematical analysis:

（1）In analysis: on the one hand, any one can use “the ‘potential infinite--actual infinite’ confusing formal language and production line” to construct all kinds of infinite related paradoxes (the Second Mathematical Crises triggered by the Berkeley Paradox has never been solved at all); on the other hand, one can also use exactly the very same “formal language and production line” to construct all kinds of infinite related “important mathematical proofs and theorems”. The typical example is: Zeno’s construction (proof) of “Achilles Can Never Chase Up Turtle Paradox” and its modern version of “creating infinite items each greater than 1/2 or 100 or 1000000000000000 or 1000000000000000000000000000000 or ... from the Un--->0 Harmonic Series and turn the Harmonic Series into a ‘Vn ---> any positive constants’ infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000)”.

（2）In set theory: on the one hand, any one can use “the formal language and production line operation of looking for some elements belonging to an infinite set but impossible to be found inside this infinite set” caused by the confusion of “potential infinite--actual infinite” to construct all kinds of infinite related paradoxes; on the other hand, one can also use exactly the very same “formal language and production line operation” to construct all kinds of infinite related “important mathematical proofs and theorems”. The typical example is: the construction of “Russell’s Paradox” and its modern versions of fundamental “Power Set Theorem proof” and “Uncountability proof of Real Number Set”

Yours,

Geng Ouyang

As you say there are many questions in Mathematics an its relation with science. Even more there is many questions about Mathematics itself.

If you look for answers, I recommend you go to the department of analytic philosophy and look for a seminar in Philosophy of Mathematics, but probably you find more questions than answers.

Studies on the infinite related mathematics history have proved that it is the inevitable fundamental defect of “’potential infinite--actual infinite’ confusing” caused by the absence of systematic theory of “scientific carriers” that decides any areas in present infinite related science (including mathematical analysis and set theory) are unable to escape from the influences and constraints of such fatal defect, unable us to conduct the quantitative cognizings scientifically to “infinite related mathematical things”. So, many infinite related paradox families (symptom complex) in present mathematical analysis and set theory have been inevitably produced. Although many ways to solve this “paradoxes symptom complex” have been invented in our infinite related mathematics (such as Standard Analysis, Nonstandard Analysis, Type Theory, Model Theory, ZFC ) since Zeno 's Achilles and Tortoise Paradox, non of them is able to solve the real trouble of “’potential infinite-actual infinite’ confusing”--------The Second Mathematical Crisis and The Third Mathematical Crisis are unsolvable at all in present infinite related science and mathematics basing on classical infinite theory system. The systematic studies and development of “the idea and theory of ‘scientific abstract law--the carriers of scientific abstract law’” (especially the systematic studies and development of “the idea and theory of ‘the carriers of science’”) will fill in a blank in the foundation of Logic Theory and open a new way and a new working field for our further studying and cognizing on Logic (the relation laws among things in the universe).

“The natural metabolism law (mechanism)” is governing the whole life process of any “alive things” in the world. Humans’ mathematics is an alive thing concomitant humans, it metabolized and evolved before, it is metabolizing and evolving now, it will metabolize and evolve in the future and, those unscientific things must be out while some scientific things must be in whenever the time is ripe. Having been suffering from the tormenting of “‘potential infinite--actual infinite’ confusing”, we should look for the new working idea and jump out of the “‘potential infinite--actual infinite’ confusing” abyss. The emergence of the new infinite theory system and the revolution in the foundation of present mathematical analysis and set theory decide the emergence of the Fourth Generation of Mathematical Analysis and the Second Generation of Set Theory. Only discarding the unscientific concepts of “potential infinite, actual infinite” in present infinite theory system and construting the new infinite theory system basing on the scientific concepts of "abstract infinite law--abstract infinite law carriers", studing and developing the systematic new "’scientific carriers--scientific carriers treating’ theories", can we fairnessly and justifiably conducting the quantitative cognitions to “infinite related mathematical things” with the systematic theory of “scientific carriers”, solving the fundamental defects disclosed by the obstinate “infinite paradoxes symptom complex” theoretically and operationally to dispel the thousands years suspended black cloud of “infinite related paradox families” over the sky of present mathematics and science

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)

I notice that you don't say where you supposedly "published" these papers. Based on your posted comments, no self respecting journal would publish them.

Thank you dear Mr. Donald Myers,

You can check from CNKI that these papers are really published with following links ：

https://www.researchgate.net/profile/Geng_Ouyang2/publications

1，On the Quantitative Cognitions to “Infinite Things” (I)

https://www.researchgate.net/publication/295912318_On_the_Quantitative_Cognitions_to_Infinite_Things_I

2，On the Quantitative Cognitions to “Infinite Things” (II)

https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II

3，On the Quantitative Cognitions to “Infinite Things” (III)

https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III

4 On the Quantitative Cognitions to “Infinite Things” (IV)

https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV

5 On the Quantitative Cognitions to “Infinite Things” (V)

https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V

You don't understand the meaning of the word "bigger" when applied to infinite sets. Two sets have the same cardinality if there is a 1-1 correspondence between them. That does not exclude the possibility that one is a proper subset of the other.

You need to consult P. Halmos book on "Set Theory" and Bertrand Russell's famous book as well as the books on logic by A. Church (Princeton Press)

Thank you dear Mr. Donald Myers,

How about “more infinite” and “less infinite”, just forget “bigger infinite” and “smaller infinite”.

Best regards!

this link may help you to analyse the 1-1 correspondance

https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/continuum/Pages/onetoone.aspx

Thank you dear Mr. Jack Son !

Combining together various infinite relating paradoxes in present classical set theory and studying them, a common characteristic among them was revealed: all of them are in fact the family members of Russell's Paradox closely relating to the mathematical things of "self and non-self, infinite and non-infinite (some elements belonging to and not belonging to a set)" with the property of T = {x|x📷x}. In present classical infinite set theory, many members of Russell's Paradox Family can be construct as long as some elements in any set are endowed with the property of " self and non-self, infinite and non-infinite" (such as those “hidden first then shown” elements with the property of T = {x|x📷x}in the proof of uncountable of real number set; Those elements with T = {x|x📷x} property in the proof of Cantor's Teorem of 📷; A barber belonging to a certain village and not belonging to that very village because of with a certain characteristic of T = {x|x📷x};...).

During the quantitative cognizing process in present classical set theory to those “not knowing-what infinite set’s elements”, people can very freely choose one of the following three “one-to-one corresponding” operations:

(1) During the “one-to-one correspondence” quantitative cognizing process to elements contained in any two infinite sets, it is a must to deny all the unique characteristics of those elements and whenever an element is given from the Set A, the Set B can always (be able to) give an element to map with and vice versa (surjection) ------- both sets are surely contain infinite many number of elements as they are both infinite sets (these elements are only “abstract points without any characteristics” except the forever nature of “infinite”, no “quantities” at all). The idea and the operation in the proof of “Infinite Rational Number Set has same quantity of elements as Natural Number Set” is a typical example.

(2) During the “one-to-one correspondence” quantitative cognizing process to elements contained in any two infinite sets, it is a must to recognize all the unique characteristics of those elements in a set (special properties, special conditions of existence, special forms, special relationships as well as very special quantitative meaning). So, it is very possible that just a tiny part of elements in the Set A map and use up all the elements in the Set B (bijection), so infinite elements from Infinite Set A are left in the very “one—to—one element mapping between Infinite Set A and Set B-------the elements’ various different characteristics within many different infinite sets inevitably decide the different quantities of elements in many different infinite sets. The ideas and operations of “Rational Number Set has more elements than Natural Number Set proof” and Cantor’s proves of “Real Number Set has more elements than Natural Number Set”, “Power Set Theorem”, “Transfinite Number Theory”, “Continuum Hypothesis” are typical cases.

(3) During the quantitative cognizing process to elements contained in an infinite set, it is a must to recognize all the unique characteristics of those elements in a set (special properties, special conditions of existence, special forms, special relationships as well as very special quantitative meaning). Hiding some elements or certain characteristics of those elements in the set at first with some means (just constructing some mathematical things with the nature of T={x|x📷x} with the idea of confusing "potential infinite--actual infinite") and doing some operations to obtain a kind of quantitative cognizing result, then magically showing out the hidden things and doing some operations to obtain another kind of quantitative cognizing result, this has been the very way to create all kinds of family members of Russell Paradox.

A fact was mercilessly revealed by those pending infinite relating paradox families in present classical set theory that since set theory came to being, people have never been able to explain scientifically at all when should or should not admit the unique elements in different infinite sets during the quantitative cognizing process to elements in different infinite sets-------any operation is self-refutation whatever former languages used.

Geng

Unfortunately (for you) everything you say is meaningless and mostly wrong. The term "Natural numbers" might have been used at one time in the distant past but you will not find it in any mathematics or logic book. You should be referencing either the set of integers or the set of positive integers or the set of non-negative integers (as contrasted with the set of Real numbers which includes as proper subsets the set of Rational numbers, the set of transcendental numbers.)

The term "infinite numbers" is nonsense

One of the important characteristics of a set with an infinite number of members is that there is a 1-to-1 correspondence between the set and a proper subset. Finite sets do not have this characteristic.

You need to recognize/acknowledge that there are different degrees of "infinity", i.e. countable, non-countable. The set of integers is countably infinite but the set of Real numbers is not countably infinite. The set of Rational numbers is countably infinite because there is at least one 1-to-1 correspondence between the set of positive integers and the set of Rational numbers

The set of even integers is countable, the set of odd integers is countable even though both of these are proper subsets of the positive integers. For each there is a 1-to-1 correspondence with the set of integers.

Your ideas about finite sets are wrong when applied to infinite sets, e.g "bigger", "smaller"

Thank you dear Mr. Donald Myers,

Many people really belive that there are "infinite", "more infinite", "more more infinite", "more more more infinite",... for different degrees of "infinite".

In fact, there are more operations:

Infinite Rational Number Set has more elements than Infinite Natural Number Set！Is anything wrong?

It is believed that there are the bijection relationships between Infinit Natural Number Set and Infinite Rational Number Set, but following simple story tells us that Infinite Rational Number Set has far more elements than that of Infinite Natural Number Set:The elements of a tiny portion of rational numbers from Infinite Rational Number Set (the sub set ： 1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) map and use up (bijective) all the numbers in Infinite Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so，infinite rational numbers (at least 2,3,4,5,6,…n,…) from Infinite Rational Number Set are left in the “one—to—one element mapping between Infinite Rational Number Set and Infinite Natural Number Set (not the integer set )------- Infinite Rational Number Set has infinite more elements than Infinite Natural Number Set.This is the truth of a one-to-one corresponding operation and its result between two infinite sets: Infinite Rational Number Set and Infinite Natural Number Set. This is the business just between the elements of two infinite sets and it can be nothing to do with the term of “proper subset, CARDINAL NUMBER, DENUMERABLE or INDENUMERABLE”.Can we have many different bijection operations (proofs) with different one-to-one corresponding results between two infinite sets? If we can, what operation and conclusion should people choose in front of two opposite results, why?”Such a question needs to be thought deeply: there are indeed all kinds of different infinite sets in mathematics, but what on earth make infinite sets different?There is only one answer: unique elements contained in different infinite sets -------the characteristics of their special properties, special conditions of existence, special forms, special relationships as well as very special quantitative meaning! However, studies have shown that, due to the lack of the whole “carriers’ theory” in the foundation of present classical infinite theory, it is impossible for mathematicians to study and cognize those unique characteristics of elements operationally and theoretically in present classical set theory. So, it is impossible to carry out effectively the quantitative cognitions to the elements in various different infinite set scientifically.

Geng

Just because some people "believe" weird things doesn't make them true. Your so-called explanation is all wrong

Thank you dear Mr. Donald Myers,

What do you mean by "different degrees of infinite" and what do you think the differences between "infinite, more infinite, more more infinite, more more more infinite,..." and "different degrees of infinite"?

Regards,

Geng

Thank you dear Mr. Jaykov Foukzon.

I agree with you that the suspended infinite related Paradoxes Families mentioned in present set theory and mathematical analysis obviously arises from the unscientific concepts of "potential infinite" and “actual infinite" as well as the abnormal postulate of the actual infinite.

Something must be done sooner or later and we are on the way to solve such thousands years old fundamental defects in human science.

Sincerely yours,

Geng

Dear Mr. Jaykov Foukzon.

But I think that the suspended infinite related Paradoxes Families mentioned in present set theory and mathematical analysis have been produced far earlier than Hilbert time.

Anyhow something must be done sooner or later to solve such thousands years old fundamental defects in human science.

Sincerely yours,

Geng

Thank you dear Mr. Jaykov Foukzon.

Sincerely yours,

Geng

Thank you dear Mr. Jaykov Foukzon.

People have been troubled by those fundamental problems such as “what is infinite and what are infinite things” for thousands of years-------we should do something new from very fundamentally.

I am now on the way to the new infinite theory system basing on the concepts of "abstract infinite and the carriers of abstract infinite"--------for all the infinite related areas (such as set theory, analysis).

We can spare no effort to develop "infinite carrier theory", and develop comprehensive and scientific cognition of various contents related to "mathematical carrier of abstract infinite concept", the contents of "abstract infinite concept (abstract scientific thing) - mathematical carrier of abstract infinite concept (the carrier of abstract scientific thing)" have been discussed and introduced.

A new concept of "Infinite Mathematical Carrier Gene" is introduced basing on the concept of "carriers of abstract infinite" to describe the basic construction and properties of infinite mathematical carriers ------- it was the born "carrier gene" that determines each "infinite mathematical carrier" has its unique characteristics, unique existence conditions and the unique form of expression, guarantees us conducting down-to-earth scientific cognitive activities (of course including quantitative cognition) to infinite mathematical carriers.

A new concept of Infinite Mathematical Carrier Measure is introduced basing on the concept of "Infinite Mathematical Carrier Gene". The new concept was created specifically for those smallest or biggest quantitative cognitive units that retain the natures of "infinite mathematical carriers”, so that we can conduct the scientific and efficient qualitative and quantitative cognizing operations to different "infinite mathematical carriers” from different perspectives---------- smallest or biggest velocity, smallest or biggest rational number, the whole quantity of elements in Natural Number Set, the whole quantity of elements in Real Number Set, different quantity of elements in different infinite sets, ...; the Infinite Mathematical Carrier Measure of elements in Natural Number Set is smaller than that of in Rational Number Set or Real Number Set because the "Natural Number Carrier Gene" determines the Natural Number Set contains only elements with "the "Natural Number Carrier Gene" but none of the other elements (such as the rational numbers or real numbers) with other “carrier gene", namely "the infinite carrier gene of rational number or real number is greater than the infinite carrier gene of natural number" determines that the infinite carrier measure of elements in Rational Number Set or Real Number Set is sure be bigger than the infinite carrier measure of elements in Natural Number Set.

Some of my papers introducing the work have been uploaded on RG.

Sincerely yours,

Geng

Jaykov Foukzon

>>>>> Contemporary logicians try to resolve this tension by using nonclassical logics

A few weeks back I funny thought ocurred to me (it made me laugh) , 'Which is the REAL naive set theory?

Regards

Jim

Your logic is all wrong, you simply don't understand and you are wasting everyone's time with your nonsense

Thank you dear Mr. Jim Moore.

People have been troubled deeply by those fundamental problems such as “what is infinite and what are infinite things” for thousands of years, so we have to do something new from very fundamentally------- nonclassical logics or developing a new infinite theory system.

Sincerely yours,

Geng

Thank you dear Mr. Donald Myers.

Would you please tell us what and how whose “logic is all wrong”?

Sincerely yours,

Geng

>>>>However consistency of the some formal set theories say ZFC meant that there exist an model MZFC of ZFC,i.e. Con(ZFC)∃MZFC

Jaykov Foukzon ’s point is well taken and I think the only way to resolve ZFC is through model theory with if not a complete model of ZFC, at least an inner model of ZFC must be established. This implies an inner for CH to include the semantic interpretations of CH.

The original question about “one-to-one” is in specific reference to the bijective Diagonal Method (DM); this cannot be separated from the ZFC notions of infinity.

The approach I have taken on the subject is to construct a minimum model of infinity from standard results. I then show that the independence of CH from ZFC distributes to all inner models of ZFC such that: (see figure)

Kind regards,

Jim