Dear Genk,

The definition of A being infinite countable is "there exist a bijection between A and  N". The definition of A being infinite uncountable is " for every function f : A ---> N, f is not a bijection".

You have recognized that there is a bijection between Q and N, so you admitted already that Q is countable. 

But you did not prove that "for every function f from Q to N, f is not a bijection". So you did not prove that Q is uncountable - and of course you cannot do this, because is countable. 

So it seems that your missunderstanding is based on missunderstanding the quantifiers "there exists", "for all", and of the sense of negation. Also, there are meta-missunderstandings, as for example that one must admit and refer to a definition, when one uses a notion. So please check those things again. 

This example does not prove that does not exist another one to one mapping (defined in another way), between Q and N.

The definition of countability is the following: A set M is countable if there exists a bijection between M and the natural numbers N.

This does not mean that the set M is uncountable if there is a bijection from a proper subset of M to the naturals.

If so, this would imply that N itself is uncountable: The map 2n -> n is obviously a bijection from the even numbers to the naturals. In your argument this would mean that N is not countable, which is obviously wrong.

Dear Geng Ouyang,

Your example does not prove that one cannot construct any "one to one" mapping from Q onto N. Usually, such a mapping "appears" by means of a related mapping, which applies "one to one" NxN onto N. This last assertion is similar to the fact that a countable joint of countable sets is countable.

Have a look at the attached link. There is a very ´visual' ordering of (positive) rational numbers. The bijection with integers is simply the position in the list.

http://www.homeschoolmath.net/teaching/rational-numbers-countable.php

Dear Professor Octav Olteanu and Professor Horst Heck, thank you for your points of view.

So, it is logical to say: rational number set is both countable and uncountable and, people can choose any decision they like？

Best New Year Wishes,

Geng

Dear Dr. Philippe Martin, thank you for your frank point of view.

But how do you think of the bijection between a tiny portion of rational numbers 1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n … from rational number set and all the numbers 1, 2, 3, 4, 5, 6, …, n … from natural number set?

It is logical to say: rational number set is both countable and uncountable and, people can choose any decision they like？

Best New Year Wishes,

Geng

Dear Professor Horst Heck,

The elements in Even Number Set is half of the Natural Number Set (otherwise it can not be called Even Number Set), but the elements in “deformed Natural Number Set constructed by 2xN = 2N” is N itself:N’.

It is of cause N ------>N itself (the N’ from deformed N).

Yours,

Geng

N is countable , M= (2k+1)  is a subsequence of N and M is countable

Geng, somebody worried once about how much infinite was an infinity, so that they created this bizarre entity called Aleph: http://mathcentral.uregina.ca/QQ/database/QQ.09.08/h/justin12.html

You are listed as my reader, so that I would expect you to have read Infinis. I think I talked about Aleph there, not sure. I definitely wrote about it before.

If they are all infinite sets, you intuitively reach the conclusion that any of them could count the others by means of correspondence one to one, is it not? We are used to count things with the natural numbers, but, on this occasion, we could decide to count with the integers if we have a set that has both negative and positive elements, right?

If they both have an infinite number of elements, it is clearly the case that we could not be short of counting elements, right?

In this way, Aleph is just a stupid thing, but somebody someday had your intuitions, which go very close to the intuitions of everyone else: One set has to be bigger than the other, thus the size of the infinity we refer to must be different, is it not?

We are however confusing counting with size of sets when we think like that: No doubts the set of the reals is bigger than the set of the natural numbers, nobody would debate over that, since the fact is obvious. 

Counting, however, means being able to associate a natural number with each element of a set. The reason why we say we cannot count the reals is because, basically, we do not know the shape of the elements in-between, like not all of them, like we ourselves have not found a way of creating all irrational numbers in between a couple of successive integers. We don't even know how to do that, how to generate all irrational numbers in-between a simple couple of successive integers, so say 2 and 3. And this is like a really tiny bit of the real numbers set.

We notice, however, that the pattern must repeat, like each couple of integers that come one after the other should follow the same pattern that we have in between 2 and 3.

It is all fascinating, I reckon. 

Basically, we cannot count them because we ourselves cannot name them, we do not know their shape to the level of exhaustion, like not even in between 0.0111111111111... and 0.011111111111...(say 2 in the end of as many as you put there). 

We would have to be able to establish a rule of construction for these irrational numbers to try to do something similar to what Cantor did. 

With the rationals, it is possible to create this rule, but not with the irrationals so far, basically. 

Dear Genk,

The definition of A being infinite countable is "there exist a bijection between A and  N". The definition of A being infinite uncountable is " for every function f : A ---> N, f is not a bijection".

You have recognized that there is a bijection between Q and N, so you admitted already that Q is countable. 

But you did not prove that "for every function f from Q to N, f is not a bijection". So you did not prove that Q is uncountable - and of course you cannot do this, because is countable. 

So it seems that your missunderstanding is based on missunderstanding the quantifiers "there exists", "for all", and of the sense of negation. Also, there are meta-missunderstandings, as for example that one must admit and refer to a definition, when one uses a notion. So please check those things again. 

I believe that diagonalisation of the proof is the missing element.

OK, William, I think I got your point. 

Geng: This proof you started to write, the one that Cantor presented, involved trying to enumerate all rational numbers. He then would write a list that seemed to be exhaustive for a certain interval, but would use a diagonal line to form a number that was for sure different from all the others, proving to all of us that not all numbers in the interval had been listed when we thought that they had, so that we had no way to enumerate, or count with order, the rational numbers. 

Dear colleagues, thank you for your frank points of view. Followings are my ideas:

Ramadan: The elements in Odd Number Set is half of the Natural Number Set (otherwise it can not be called Odd Number Set), but the mapped elements in “Odd Number Set of deformed Natural Number Set constructed by 2xN+1 = Odd Number” is N itself--------on the way Odd Number but not really Odd Number. It is of cause N ------ > N itself (the N’ from deformed N).

Marcia: 1, the intuitions of smaller or bigger is unavoidable and very important in our mathematics when doing numerical cognizing to anything including infinite-------on the directions of small and big. Our history tells us that we have a lot achievements in the infinite related numerical cognizing field (such as in analysis and set theory).

2, No doubts Real Number Set is bigger than Natural Number Set because we all know the R has more elements than the N. with my story at the beginning of this thread we notice there is the fact that there are more elements in Rational Number Set than that in Natural Number Set, why not “No doubts Rational Number Set is bigger than Natural Number Set”?

3, I really agree with you that we must establish a rule of construction for these irrational numbers, but this is another task.

Mihai: if there is the fact that there are  far more elements in Rational Number Set than that in Natural Number Set, we get a conclusion: “Rational Number Set is bigger than Natural Number Set”. And this is enough.

William: I agree with you.

Yours,

Geng

Dear Geng, finite, infinite, countable, non-countable, bigger, smaller... only words without any meaning if we does not use a frame in which preciselly determined them. For example both of the set theory and measure theory the fact that "there are more elements in Rational Number Set than that in Natural Number Set" does not means that "Rational Number Set is bigger than Natural Number Set" but with respect to an "everyday language" this conclusion is right. The question is: A determined meaning of a word is good for you or not  suitable to express your intuition? In mathematics, you may choose to give another meaning of the words which currently used, if you are right (in this) your definition will displacing the currently used one in the future. 

So give us more usable concepts rather then the next: finite cardinality, infinite cardinality, countable or non-countable sets ... I think that the community of mathematics sufficiently open to consider the new definitions which help us to understand our world.

There is noththing to say more.

In my opinion Mihai has explained everything better than nice.

Q is countable.

Regards, Tadeusz

Dear Professor Á. G. Horváth, thank you ever so much for your kind and insightful point of view in your post: “The question is: A determined meaning of a word is good for you or not suitable to express your intuition? In mathematics, you may choose to give another meaning of the words which currently used, if you are right (in this) your definition will displacing the currently used one in the future.”, “So give us more usable concepts rather then the next: finite cardinality, infinite cardinality, countable or non-countable sets ... I think that the community of mathematics sufficiently open to consider the new definitions which help us to understand our world”.

It is enough for me if as you put it: “with respect to an ‘everyday language’ this conclusion is right.” Because I am very sure that this common intuition "everyday language" is not far from "mathematical language"

As you and Professor. Mihai Prunescu may know, basing on the intuition as Professor Marcia Ricci Pinheiro put it,  I am now a single fighter working very hard to integrate those confusing and “trouble making” different definitions of infinite (such as "actual infinite" and "potential infinite") in our science ---------the task basing on the achievements people have done in our thousand years science history and standing on the giants shoulders. This working field is far more than mathematics.

I am now busy in discovering many defects and their forms in the foundation of present classical infinite related science system that people in many fields (such as in philosophy, linguistic, physics, …) can understand and someday in the future the new cognizing fruits can be naturally put into philosophical language, linguistic language, physical language, mathematical language,… On the other hand, I really don’t have enough time to do all the “decorations”.

My heartfelt thanks to you, Dear Professor Á. G. Horváth.

Best New Year Wishes,

Geng

Dear Geng Ouyang, I believe that you were puzled somewhere in hard research. I did not read Professor's Horváth in detail but clearly that in maths you must be careful,the terms, being precise. You might have mixed up sets inclusion, clearly that the set of natural numbers is just a part of the set of the rationa numbers. With the concept of cardibality of a set that is, roughly speaking, two sets have the same cardinality whenever there is a bijection between them. You may go on counting indefinitely either 1, 2. 3, ... or 0, 1, 2, 3, ....these two sets have the same cardinality, for you put them through an injection that clearly also is onto, hence a bijection, The same holds for the cartesian product N(0) x N(0) where N(0) stands for {0, 1, 2, ...}. You easily find a way. Now you may find a bijection betweren N(0) x N(0) and the set of the rational numbers, see? You just recall that both N and N(0), Q are what you use to say, countably infinite sets.

Dear Professor Nuno Freire and Professor Miguel Ángel Montes, thank you very much!

Nuno: Many RG friends are right that I am now really working in a “strange prime intuition way” which is different from many people.

I had a very sad and trouble beginning in the first period of my study on the “very undoubtedly strict mathematical divergence proof of Harmonic Series”. As time being, I found the newly discovered Harmonic Series Paradox is in fact a member of 2500 years old Zeon’s Paradoxes family and more troubles came because I found myself in an unbelievable huge working field--------those defects disclosed by suspended Zeon’s Paradoxes family are there, they should be solved by us own.

My conclusion is: the “2500 years old huge black cloud of infinite related paradoxes over mathematics sky” is produced by too less intuition and ontology but too much formalism in present classical infinite related science system-----an integrating work is badly needed. So I have following 3 working principles:

1, pick up “prime intuition” in my studies on those defects disclosed by suspended Zeon’s Paradoxes family in present classical infinite related science system.

2, basing on the new achievements from “prime intuition of human science”, not to be “polluted” by some “terms and results” relating to those discovered defects disclosed by suspended Zeon’s Paradoxes family in present classical infinite related science system.

3, the new achievements from “prime intuition of human science” are right there unavoidable (a revolution in “infinite”, “infinite related number system”, “infinite related limit theory”); they will be further developed someday by some people from different fields.

Miguel: for "Set A is bigger than Set B", I mean “Set A has more elements than Set B”. According to my “strange prime intuition way”, I think we have being neglecting too much on the “definition of set” but paying too much on the “form of set”. According to the “definition of set”, not only rational number set is bigger than natural number set, but also natural number set is bigger than even number set and odd number set------this is the really “practical infinite”（actual infinite）but not “jumping up and down as one wishes on the confusing actual infinite and potential infinite”.

Best New Year Wishes,

Geng

Dear Professor Miguel Ángel Montes, thank you very much for your kindness!

How do you think of following story (forgetting about “bigger” or “uncountable”):

Just take a tiny portion of rational numbers (elements) from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) and they map and use up (bijective) all the numbers in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so a lot of rational numbers (elements) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set”.

 So, Rational Number Set has more elements than Natural Number Set.

Best regards,

Geng

Dear Professor Miguel Ángel Montes, thank you very much for your frank point of view.

I have a frank point: my work now is far more than “infinite in a point of view used during all european medieval age and before”.

1, the “divergence proof of Harmonic Series” to me is a paradox produced by fundamental defects of classical “infinite”, “infinite related number system”, “infinite related limit theory” in present classical infinite related science system, but to all european medieval age and before is a “very undoubtedly strict mathematical proof” supported by solid foundation of classical “infinite”, “infinite related number system”, “infinite related limit theory” in present classical infinite related science system.

2, many clear and unavoidable defects disclosed by the mistakes and paradoxes in present classical infinite related science system to all european medieval age and before are unrelated and nothing to do with each other but to me these defects are “blood brothers and sisters” integrated very closely.

Best regards,

Geng

Aytekin, I reckon you wrote one thing and expressed another on your first line there. Please reassess. 

My mistake. The right one is that 'If there is a one-to-one function from N to R, the cardinality of N is less than or the same as the cardinality of R and we can write |N| ≤ |R|.'

The set of rational numbers is countable!

Dear Professor Mohsen Aliabadi, thank you.

How do you think of following story (forgetting about “bigger” or “uncountable”):

Just take a tiny portion of rational numbers (elements) from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) and they map very well to all the numbers (elements) in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so a lot of rational numbers (elements) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set”.

 So, Rational Number Set has far more elements than Natural Number Set.

Best regards,

Geng

Dear Geng

Lets use your own reasoning in a different way:

Just take a tiny portion of the Natural Number Set (2,4,6, ... ,2n, ...) and they map very well on the set of all Natural Numbers  (1,2,3,...,n,...). So a lot of natural numbers are left behind in this one -to-one mapping (2n onto n) from the Natural Numbers onto the Natural Numbers, Hence your conclusion should be: the Natural Number Set has far more elements than the Natural Number set. Think about this.

Best regards, Joseph.

 Dear Geng Ouyang, by using your idea, the function f(x)= tanx map the interval             (-π/2, π/2) very well to R the set of real numbers, 

so a lot of real numbers (elements) from the interval (-2π,2π) are left in the “one—to—one element mapping between (-π/2, π/2) and R the set of real numbers.

 So, the interval (-2π,2π) has far more elements than R the set of real numbers.

I think you need to reed equipotent of sete and cardinal numbers.

 Dear Geng, 

I think the answer of Professor Joseph Gielen should convince you. Of course, in spite of having the same cardinality, these two sets of numbers, have the different structures;

For instance in the set of natural numbers we have the notion of successive numbers

but in the set of rationals this notion does not mean. The set of rational numbers is a dense subset of the set of real numbers, but natural numbers is not so and ...

Best regards,

Jafari.

Dear Professor Joseph Gielen, Professor Abdalla Ahmad Tallafha, Professor Saeid Jafari and Professor Baravan A. Asaad, thank you so much for your frank points of view. The following is my point about subsets, let’s take Even Number Set and Odd Number Set as examples:

According to the definition, the original Even Number Set and Odd Number Set is each half of the Natural Number Set (otherwise it can not be called Even Number Set and Odd Number Set). People have been using two deformed Natural Number Set of (1x2, 2x2, 3x2,...,nx2,...) and (1x2+1, 2x2+1, 3x2+1,...,nx2+1,...) and used all the natural number components (but not the whole thing of “2,4,6,8…2n” and “1,3,5,7,9…nx2+1”) to map natural numbers, of cause they map very well because it is N ----> N itself but not 2n ----> N and nx2+1----> N.

What are mapping to natural numbers: the even numbers and odd numbers or the components of even numbers and odd numbers?!

I am very sorry to say that Dr. Joseph Gielen and I have totally different way of mapping for Natural Number Set and Even Number Set. So, his idea has nothing to do with my conclusion.

In present classical infinite science system, people have been paying too much attention to the forms but too less to the ontology at less since Zeon’s time.

Best regards,

Geng

I think we should first defined what the meaning of bigger!!!!!

We should stay in definition and then talk about big or small. 

If X bigger than Y means car(Y)

When we talk about the sets with infinite cardinality then there is always problem due to our lack of knowledge about infinity. In fact in my opinion there are infinite type of infinity. Up till now we are able to classify few of them. I hope in future we will be able to improve our knowledge of infinity.

If we assume the concept of 'X is bigger than Y' in the sense of cardinality "Cardinality(X)

Dear Professor Yaser Maleki, Professor Ismat Beg and Professor Siginam Sambasivarao, thank you very much for your view points.

1, is it right we say "Set A is bigger than Set B because Set A has more elements than Set B"?

2, I think in our science we can have only one definition of infinite but we can have infinite type of “infinite carriers”. So, we have “theoretical infinite” and “applied infinite”. In fact, the suspended infinite related paradoxes disclosed a truth: for more than 2500 years, we human have done successfully a lot in the field of “applied infinite” but not enough on “applied infinite”.

Best regards,

Geng

Dear Gen Ouyang,

First of all, if we talk about bigger of two sets, definitely it is related to the cardinality.  If both sets are finite, then it is easy to compare the cardinality from school maths. But, if both sets are infinite, then cardinality theory helps to classify those sets.

Obviously N and Q are different from each other. Otherwise why there are two different names for them? But once we come to the point of countable, they belongs to the classification of countable sets. This is some thing like people are disintegrated as americans, africans, asians europeans etc, but when they are all classified as human, they come under the same classification.

There are a few concepts which we have to understand. Let us consider the set of Natural numbers as N, the set of Integers as Z, the set of Rational numbers as Q and the set of real numbers as R. Then obviously, N subset of Z subset of Q subset of R. Therefore, It looks like Q has more elements than N and hence Q looks like bigger than N (if we look at them by wearing subset spectacle and finite spectacle). But how much more? Since both are infinite sets, it is difficult to judge that immediately, because if we add/remove infinite quantity of elements to/from an infinite set still it can be infinite. There is a concept called equivalence. There is a difference between equal and equivalence. Two sets A and B are equal if and only if A subset of B and B subset of A, whereas A and B are said to be equivalent if there exists a bijective map between A and B. If two sets are equal, they are equivalent as well, but the converse is not true. If two sets are equivalent, then they must have same cardinality. Obviously N and Q are not equal, but they are equivalent. In terms of infinite theory, there are different types of cardinality, aleph-null, cardinality of continum, etc. In this sense, N, Z, Q and many more belongs to aleph-null class. N, Z, Q are equipotent, equinumerous or equipollent sets. That means their cardinality is same. So, if we wear this class, Q is not bigger than N or N is not bigger than Q.

A set is said to be countable, if it can be enumerated as a sequence. Also, a theorem on countable theory says that countable union of countable set is countable. You have given a map which maps Q1 = (1, 1/2, 1/3, 1/4, ..1/n ...) to N=(1,2,3,..,n..). I can show another map which maps -Q1 to N. If we take -Q1 U Q1 , then it is a subset of Q and it is countable. If you can give as many maps as possible, it is possible to take their union and prove their union is countable, by above mentioned theorem. Definitely, it is impossible to provide uncountable maps from Q to N.

If you still don't agree, would you say N U {0} , Z are bigger than N? If so, could you prove |Z| > |N| or |N U {0} | > N?

Also, as per one of your comment, 2N and N are not equal, so N is bigger than 2N, but they are not,  because their cardinality is same aleph-null.

Dear colleagues,

TThe elements of tiny portion of rational numbers from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) map and use up (bijective) all the numbers in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so infinite rational numbers (at least 2,3,4,5,6,…n,…) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set”.

Is it meaningless if "Infinite Set A has infinite more elements than Infinite Set B"-------- Rational Number Set has infinite more elements than Natural Number Set (forgetting about “bigger” or “uncountable” first)?

It is meaningful. But, mathematicians have defined terminologies in terms of equivalence class and certain classifications to study about their properties. I agree that rational number set has infinite more elements than natural number set. Of course, prime number set is an infinite set, but natural number set has more infinite elements than prime number set. So, inside natural number set itself, you can find infinite number of infinite subsets, where each subset has infinite elements than another set. For example, if we consider 4N and 2N, 2N set has infinite more elements than 4N. However, they are all countable set and hence their cardinality is same.

Dear Professor Panchatcharam Mariappan,

The elements of tiny portion of rational numbers from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) map and use up (bijective) all the numbers in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so infinite rational numbers (at least 2,3,4,5,6,…n,…) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set”------- Rational Number Set and Natural Number Set are not bijective at all.

What decides the same cardinality of two sets?

Best regards,

Geng

Although, your mapping has some numbers left with, we can find another function between N and the left over number, which would prove that both mapped and left over are countable. Therefore their union is countable. Cardinality or Cardinal number is defined and decided using bijective functions only. https://en.wikipedia.org/wiki/Cardinal_number

Dear Professor Panchatcharam Mariappan,

If The elements of tiny portion of rational numbers from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …)  map and use up (bijective) all the numbers in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …), and infinite rational numbers (at least 2,3,4,5,6,…n,…) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set”, how could you operate exactly that “we can find another function between N and the left over number, which would prove that both mapped and left over are countable” as you put it in your post?

Regards,

Geng

Ok. Let me explain that in an elmentary level. Consider x2 + 9x + 18 = 0. In this case, I can say that x = -3 is one solution. So, as per your argument, I could say that, x = -3 is satisfied, so I don't need to consider any other possibility. But, mathematics and mathematicians are not satisfied with a single solution. They are finding all possible answers for this equation to solve. So, obviously x = -6 is another solution.

In other words, if a 100 yuan can be given as a single note or it can be given as 100 single yuan or 1000 jiao or 10000 fen,  I hope you will accept all these coins as 100 yuan currency. If you consider yuan as one map, jiao as second map, fen as third map, they are all different. When a single currency has three different maps, why are you always choosing a particular map?

Now, let us come to the point of countability. From Natural number to Rational number, there are infinite bijective maps available. You have found a bijective function between N and a subset of Q (say jiao map), it does not mean that that is the one and only bijective function. As per the countability, it says that at least one bijective function should exist between N and Q in order to Q to be countable. In otherwords, you can say that Q is not countable, if there is no possibility to find a bijective map between Q and N. But, that is not true, because there are many other possibilities available.

If I could explain to a common man, to reach our home from our office, there may be many possible maps to reach. Let us consider that you chose a particular way (map) to reach home. It does not mean that there is no other way to reach your home from your office. Ultimately, the aim is to reach home. Also, it is not necessary that the map you have chosen to reach your home is optimal due to traffic regulations. Another person can choose an optimal map to reach your home.

Again, if a way is blocked on our regular way to reach home , will we stop reaching home. Won't we find another possible way to reach our home or will we stop at the blocking and sleep there? No one would do that. Because man knows how to get rid of obstacles. The function, which you have chosen puts an obstacle to count all numbers of Q using N. That means, the function provided by you (or the way you have chosen ) is not qualified to count Q (to reach home). Choose different map to reach home (or count Q). Fortunately, our ancestors have discovered many such maps. So, Q is countable and hence N and Q are is same equivalence class.

To prove left over to be countable, again you have to find a map between N and left over. Also, there are proved results and theorem, which say that subset of a countable set is countable. Since, you are worried about the set  say Q2= {2,3,4,...,n,...}, which is a subset of N. By theorem, it is countable. Already, you have proved that Q1= {1, 1/2, 1/3, 1/4, ... 1/n,... } is countable. So, now we can prove that Q1 U Q2 is countable. (Theorem 1: Since countable union of countable set is countable). For the left over, you can argue in the same way. For example, among the left over, Q3={1/2, 2/3, 4/5, 5/6, ... n/(n+1)...} is a subset which can mapped by n -> n/(n+1). If we continue this process for each left over, you can prove, each set is countable and again using therom1, Q = Un Qn is countable.

Dear Mr. Panchatcharam Mariappan, thank you!

Many RG friends are right that I am now really working in a “strange prime intuition way not believing some long-known reasons (such as some theorems)” because of so many members of suspended infinite related paradox families.

1, there is only one Rational Number Set with its definition and one Natural Number Set with its definition in the bijection proof.

2, I agree with you that there may be many bijective ways between Rational Number Set and Natural Number Set; can one choose any way he like freely as mathematical basic theory? Why?

3, if “by theorems” can really solve all the troubles as many people believe, the divergence of harmonic series is a “strict proven mathematical basic theory”, but how do you think of following case:

The following proof is very elementary but important.

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                    （1）

=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．     （２）

>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．               （3）

=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                             （4）

Each operation in this proof is really unassailable within present science theory system. But, it is right with present modern “theorems” and technology applied in this proof that we meet a “strict mathematical proven” modern version of Ancient Zeno’s Achilles--Turtle Race Paradox: the “brackets-placing rule" decided by limit theory in this proof corresponds to Achilles in Zeno’s Paradox and the infinite items in Harmonic Series corresponds to those steps of the tortoise in the Paradox. So, not matter how fast Achilles can run and how long the distance Achilles has run in “Achilles--Turtle Race”, there would be infinite Turtle steps awaiting for Achilles to chase and endless distance for him to cover, so it is of cause impossible for Achilles to catch up with the Turtle; while in this acknowledged modern divergent proof of Harmonic Series, not matter how big the number will be gained by the “brackets-placing rule" (such as Un’ >10000000000) and how many items in Harmonic Series are consumed in the number getting process by the “brackets-placing rule", there will still be infinite Un--->0 items in Harmonic Series awaiting for the “brackets-placing rule" to produce infinite items each bigger than any positive constants, so people can really produce infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 10000000000), “strictly proving” that Harmonic Series is divergent. In so doing, the conclusion of “infinite numbers each bigger than any positive constants” can be produced from the Un--->0 items in Harmonic Series by brackets-placing rule and Harmonic Series is divergent" has been confirmed as a truth and a unimpeachable basic theory in our science (mathematics) while “the statement of Achilles will never catch up with the Turtle in the race” in Ancient Zeno’s Achilles--Turtle Race Paradox has been confirmed as a “strict mathematical proven” truth and a unimpeachable theorem------it would be Great Zeno’s Theorem but not Suspended Zeno’s Paradox! ?

My heartfelt thanks to you again dear Mr. Panchatcharam Mariappan.

Yous,

Geng

Dear Geng Ouyang,

I am not sure why you mention paradoxes in those two examples. The harmonic series is divergent since it dominates a series which is itself divergent; Zeno's paradox was a paradox only before the time when the notion of convergent series was understood, since at that time it was puzzling that a sum of infinitely many terms could be finite. What do you mean by "Great Zeno’s Theorem" and "Suspended Zeno’s Paradox"?

Dear Philippe Martin, thank you.

1, I agree with you that it has been believed at least for 1000 years that “the harmonic series is divergent”. But I don’t believe because no one in the world now can answer this question: how many items each bigger than 1/2, or 100, or 1000000, or 10000000000,… can be produced by the “brackets-placing rule" from Harmonic Series, why?

2, by "Great Zeno’s Theorem" I mean: knowing the defects in present classical infinite related science system, ancient Zeno proved beautifully that Achilles can never catch up with the Turtle in “Achilles--Turtle Race”-------modern people can produce infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000,…by the “brackets-placing rule" from Harmonic Series.

3, by "Suspended Zeno’s Paradox" I mean Zeno’s Paradox families have been there changeling us human’s wisdom for at least 2500 years.

Yous,

Geng

Dear Geng

As I said earlier, I am repeating again. N, Z, Q are in an equivalence class called countable set. The definition of countable set says that at least one bijective function should exist between N and Q. (For example, if an university requires at least one professor to teach mathematics, they put advertisement and about 100 persons would apply for the same post with certain skills. It does not mean that all 100 persons would be qualified for the post. As per university rules and regulations, they are not allowed to select a candidate freely as they wish, they would select one or two potential candidate as they required. The function you have chosen is not a potential candidate for countable set as it is not mapping the whole Q to N. So, the university(definition) disqualifies the person(function). If all 100 candidates are disqualified, university may advertise and search again. If they came to a conclusion that no such candidate exist in the world, then as you said we can argue that Q is not countable. But there are many potential candidate exists in the world who map Q to N bjiectively).

Again, the definition of countable set is already well-defined. That clearly says that at least one bijective function should exist. If you want to change at least to for all or your own definition, you are allowed to define and name it as Geng's countability or whatever you like (except the standard countability) and allowed to develop an equivalence class as you wish. No one could stop you to define that. For example, infinity is not part of real number or complex number in general, but Riemann Sphere has infinity as a member. In the same way, you can define your own equivalence class and  your own definition. If it is well-defined, it may lead to a new theory in future.

can one choose any way he like freely as mathematical basic theory? Why?

Yes. One can choose, but a theory should evolve from your own freely defined definitions. For example, in school days 1 + 1 = 2, 1 + 1 + 1 = 3, but once we started to learn algebra in higher studies, the + has a meaning. + is just a binary operation, Depending on the definition of the binary operation, the result changes. For example, 1 + 1 = 0 and 1 + 1 + 1 = 1 when + is a modulo division of 2, similarly 1 + 1 =2 and 1 + 1 + 1 = 0 for the modulo division of 3. So, which one is the correct answer for 1 + 1? Depending on the relations there are already many mathematical theories evolved. So, as I said earlier, you can define your own mathematical theory and create your own universe. But, it does not mean that an already existing universe should be embedded in your universe or vice versa. Your theory may or may not fit into the standard theory.

The Harmonic Series which you have mentioned is a divergence series but diverges slowly. If you use calculus to find the value of this harmonic series, you would get log n - log 1 which is divergent as n tends to infinity, but diverges slowly.

Related to Zeno's paradox: As Phlippe Martin mentioned it was before convergent series was really understood well. It was believed that earth is flat, sun revolves earth etc in ancient times. A well developed theory was not available at the time of Zeno's. Now, a well established theory provides a solution for his paradox. The problem which is unsolved or has no explanation would have an explanation in the future. That is basic theory of science and nature.

We should follow the definition

what is the difinition of biger set?

No such definition for infinite seats !!!!

Mr. Panchatcharam Mariappan,

1,  in the very bijection proof we talk about, there is only one Rational Number Set with its definition and one Natural Number Set with its definition.

2, one apple + one apple = two apples.

3, how many items each bigger than 1/2, or 100, or 1000000, or 10000000000,… can be produced by the “brackets-placing rule" from Harmonic Series, why?

Mr. Ramadan Sabra, the trouble is: in present classical infinite science system, many people (including Cantor) have been switching between “potential infinite” and “actual infinite” in practical infinite related mathematical operations (applied infinite) and paying too much attention to the forms but too less to the ontology (theoretical infinite) at least since Zeno’s time.

Thank you!

Yous,

Geng

Joseph, I think you confuse cardinality with size of sets. Cardinality is infinity, but the size is different. It is obviously the case that 2N is smaller than N. Maybe we have not yet created a way to refer to this, even though the thing about the Alephs was a good attempt. 

Geng: Cantor, not Kantor. Zeno, not Zeon. 

Miguel, Mathematics is perhaps ALSO about intuition: Why do we classify that as infinite? Basically because we tried to physically count it in our heads and saw that there was no end. We first did that to then come up with the idea of representing things through a horizontal 8 and etc.

Dear Dr. Miguel Ángel Montes  and Dr. Marcia Ricci Pinheiro, thank you very much!

I agree with Dr. Marcia Ricci Pinheiro that the mathematics we are now working on is human’s (not ant’s, fish’s,…), so human’s intuition is really very important. It is very difficult (impossible?) to have “the arguments of infinite related foundation of science” not related to philosophical ideas, just because all the arguments touch the root of mathematics and this is a huge reforming working field.

Dear Dr. Marcia Ricci Pinheiro, I have corrected the mistakes in the post, my heartfelt thanks to you!

Best regards,

Geng

I actually do have the formal title, Geng, so that you could say Dr. or PhD. That feels better, I reckon. Some call me Prof. for long.

Marcia, "and etc." is redundant.

Dear Geng

Simply asking, could you answer the following. As per your argument, which of the following is bigger? X = N U -7N, Y = NU -11N, Z = N U -13N. You can't answer for this. Please read further

Suppose every one agrees that Rational number set is bigger than Natural number set. Let us say N has infinite number of elements = infinite1. Then a person comes and says let me join {0} and N, then as per your argument W= N U {0} is bigger than N, because, I can map, N -> N and left with 0, so as per your argument W has another cardinality say infinite2. A second person comes and adds -1 to W and he gets W1=W U {-1}, by repeating the same argument, we would end up with infinite2. If I keep on repeating the same argument, we would get infinite3, inifinite4.... Now what will be the argument for Z, Z\{0}, N U -7N. N U -3N etc? How many names would you give. Suppose you have given a name for each some how. I have further question. Which set is bigger among the following X = NU -7N or Y = N U -3N? It is impossible to find a relation like X subset of Y or Y subset of X. How are you going to argue for this? There are infinite number of sets like N U -pN  available, where p is a prime number  > 2. If you could distinguish these two sets with a notion of cardinality, we can think of bigger and smaller. For each infinite set combined with N, how many names would you give?

Dear Geng,

 from the posts you wrote it seems you can't convince yourself of the very existence and consistency, within the mathematical world, of infinite sets. You should re-read the basic definitions of the set theory (developed starting around 1870).

Notice that  "is bigger than" means "contains" however a bijection between a set which contains another strictly is possible : whenever one such map exists, the sets have the same cardinality.

Other point : the infinite series

 SIGMA_{k=1}^(+Infinity) 1/k = +Infinity

or 

SIGMA_{k=1}^(+Infinity) 1/k^2 =pi^2/6

  written with an ifinite summation use actual (countable) infinite sums, while the writing

lim_{N -> +Infinity} SIGMA_{k=1}^(N) 1/k = +Infinity

 is more interpretable as a potential infinity.

Today these forms are equivalent, and there is no issue thinking in terms of actual infinite in the standard analysis (and even in the non-stantard one ?). The pending point is the existence or not of an intermediate cardinality between this of N and this R, the set of the reals (hypothesis of the continuum Cantor 1878, 23th Hilbert's problem 1900).

Personnally I am intuitively sure that there is no intermediate wetween a countable infinite set and the cardinal of the real numbers. But this is just a belief!

Yet you can't ask a human to write or count all the terms in an infinite sum : you have to take it for existing really, both in the mathematical world and in the world of ideas.

I would even add that the deep works of Cantor, Dedekind, Zermelo, Weierstrass, Gödel, Cohen on these topics made the subject clear, much more than the philosophical hesitations.

I fully agree with Mr. Panchatcharam Mariappan that Zeno's paradox is not a paradox anymore : it's nowadays a very useful example for a teacher to introduce infinite sum.

Hope this will help, 

Best Regards,

 Gabriel

Dear Dr. Panchatcharam Mariappan  and Dr. Gabriel Thomas, thank you very much!

Panchatcharam: there is only one  Natural Number Set with its definition in the very bijection proof we talk about, if you say “let me join {0} and N”, we can only have N U {0}-----> N U {0}, nothing left at all.

Gabriel:

1, well, we may not say “Rational Number Set is bigger than Natural Number Set” but “Rational Number Set has infinite more elements than Natural Number Set and there is no bijection relationship between Rational Number Set and Natural Number Set at all”.

2, we may forget Zeno's Paradox for a while but try to find an answer to following question:

how many items each bigger than 1/2, or 100, or 1000000, or 10000000000,… can be produced by the “brackets-placing rule" from Harmonic Series, why?

Best Regards,

Geng

It is the converse of what I said. Rational Numbers have the same count of elements as Natural Numbers : this is against common sense, but right for infinite sets with this cardinality.

B. R.

Dear Dr. Gabriel Thomas, thank you.

How do you think of following two questions (your own understandings but not other’s “common sense”):

1, The elements of tiny portion of rational numbers from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …)  map and use up (bijective) all the numbers in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …)-------- are there infinite rational numbers (at least 2,3,4,5,6,…n,…) from Rational Number Set left in the “one—to—one element mapping between Rational Number Set and Natural Number Set” so there is no bijection relationship between Rational Number Set and Natural Number Set at all (forgetting about “bigger--smaller”, ” cardinality” and “countable--uncountable”)?

2, How many items each bigger than 1/2, or 100, or 1000000, or 10000000000,… can be produced by the “brackets-placing rule" from Harmonic Series in present science theoretical frame, why?

Best Regards,

Geng

A basic set theory text will define one-one onto pairings between the natural numbers and the rationals. Generally, the positive rationals are arranged in an infinite array with numerators determining column and denominators determining row.  A diagonal enumeration of the elements of the array [skipping repeated values] gives the desired pairing -- ie., 1/1, 1/2, 2/1, 1/3, *, 3/1, 1/4, 2/3, 3/2, 4/1,..etc. Notice that every positive rational with  numerator+denominator

Dear Dr. William Richard Stark, thank you very much.

It is true that we really can see the very diagonal proof in many books.

But is there anything wrong in my above post and why?

Looking forward to your frank view points, thank you.

Best Regards,

Geng

Dear Geng

You can't compare two countably infinite set as they are all in same equivalence class. Till now you did not answer my question. Let X = N U -7N and Y = NU -11 N. Which one is bigger X or Y?

Why did you map conveniently N U {0} --> N U {0}? Always, you are talking about N and a subset of Q. Then I map N and a subset of N U {0} say N and left with {0}. Earlier you said why there is no choice to use bijection freely. I am arguing in the same way. Now I have chosen a bijetion freely. As per the same argument you can say at least one element more than N and hence NU{0} is not countable. In the same way, N U -7N has infinite more elements than N. I can construct infinite such sets. Would you give a name for each of them or name them as not countable?

Let us assume that Q has infinite more elements than N. Let EQ = Q U { e, e2, e3, ...} Would you say EQ has infinite more elements than Q. But, this EQ is also a countable set.

For the harmonic series, we can produce infinite such bracket-placing rule. Mathematics is an abstract subject. So, it is not necessary to put bracket placing rule for each. We can generalize some concepts by keeping certain conditions. Infinite is like a ocean where water will be there always irrespective of any action. You may get tired of bracket-placing rule, it does not mean that there no more. Whatever number you give, I can give a bracket-placing rule. Again, at the time of Zeno's, abstract theory was not well developed. Unsolved problem of a century need not be unsolved for ever.

In order to understand this, read more theory about epsilon - delta, continuity and measure theory. Definitely you will understand the beauty of abstract concepts.

 Dear Dr. Panchatcharam Mariappan, thank you.

1, for me, one thing is very sure and clear from very beginning for my responses to all the posts here: the focus in this bijection case is the “original” Rational Number Set and Natural Number Set. It might be good to start a new topic for “Let X = N U -7N and Y = NU -11 N. Which one is bigger X or Y”.

2, I am afraid that you may not be aware of a fact: it is the foundationless “epsilon -- delta” related limit idea and theory in present classical infinite related science system that produces Paradox of Harmonic Series.

Best regards,

Geng

Dear Geng.

1. Answer to this question, lead to the same answer to your problem. In mathematics, instead of solving a problem, we can go step by step. There are plenty of such X and Y, before moving to the set of all integers Z. How many such new topic would we discuss? It is already discussed and hence equivalence class countable set is created. 

2. If epsilon-delta definition goes wrong, most of the physics and engineering problems will go wrong. For, instance tan x -> infinity when x -> pi/2. Even modern computer and approximation would fail. Because, most of computer related real world application is constructed based on Q is dense in R.

It is something similar to showing a zero object. It is not possible to show zero object to any one and infinity object to any one. That does not mean that whole mathematics is absurd.

https://plus.maths.org/content/does-infinity-exist

Dear Dr. Panchatcharam Mariappan, thank you.

1, for me, one thing is very sure and clear from the very beginning for my responses to all the posts here: the focus in this bijection case happen between the “original” Rational Number Set and “original” Natural Number Set, and there is only one “original” Rational Number Set with its definition and one “original” Natural Number Set with its definition here------forgetting about any deform Natural Number Set or any deform Rational Number Set.

It might be good to start a new topic for “Let X = N U -7N and Y = NU -11 N. Which one is bigger X or Y”.

2, Limit theory is a very useful cognizing tool in human science. It treats “infinite number forms”. But its foundation is empty:

(1), what kind of “infinite number forms” it treats-------potential infinite number or actual infinite number or missing ups or …?

(2), in present mathematical number system, there are no “infinite number forms” at all.

I have put up a new topic: How can we make up the defects in the foundation of limit theory？

Best regards,

Geng

Geng, because of one of  Zeno's Paradoxes,  The Dichotomy, the one I solved, as you probably saw here, on my RG profile, I actually got to think more about limits. There is a fundamental detail that has been overlooked, I reckon: Sometimes we get exactly to the limiting point, as in limit of y when y=x and x goes to 1. Sometimes we never get to it, as in limit of y when y =1/x and x goes to infinity. We are not really representing that difference with symbols so far. 

Dear Dr. Marcia Ricci Pinheiro, thank you.

I think many people will agree with you that “Sometimes we get exactly to the limiting point, as in limit of y when y=x and x goes to 1. Sometimes we never get to it, as in limit of y when y =1/x and x goes to infinity. We are not really representing that difference with symbols so far.”

Would you please tell us briefly here how you solved Zeno’s Paradoxes, The Dichotomy?

Best regards,

Geng

Sure, Geng: I work a lot with Language and Logic, as you will notice, Classical Logic. 

Basically, what we see in Zeno's Paradox is a confusion between the sigmatoid half when acquiring a non-mathematical sense and the sigmatoid half when acquiring a mathematical sense. When we use half in normal language, we obviously would have to refer to the world object visual half, that is, half of what our eyes can see, right? Even because we would not fit a point in the ruler of the size we need to fit in order to take the proposed step and stop right over half of the mathematical interval. Furthermore, we will find difficulties with marking the mathematical interval in real life and then walk over it even if we were a computer dot. 

 Even in a computer, given the way it is built, the dot will eventually reach the proposed destination, since the dot ALSO suffers from all the problems our body suffers: It is bigger than the mathematical entity that we would simply be lucky if ever succeeding in finding exactly on an axis. The dot will reach the destination because our speech, when saying that, regards our visual half, not the half of Mathematics, which would have to be abstract and only seen in our heads as something precise or, at most, in theory, but not in practice.  

I actually used Cathy Freeman (sorry if I misspell now) as an example. I said that if someone were dying and she had to reach the boot to make a 000 call, but she decided to go in the way Zeno's proposed, she would definitely reach the boot, despite. First of all, we observe that our feet will occupy way more than the needed ruler dot, and therefore we will definitely reach the destination 'in human language'. 

Mathematics does go with Computer Science most of the time, but, on this occasion, we would have to agree that there is always a maximum amount of detail that the computer/program allows for, so say it can deal with 10 to the -9 in decimals. When that decimal maximum has been reached, the computer will say that the dot did reach the target, so that it will reach the target anywhere where we choose to apply the problem.

The little computer dot and Cathy cannot reach the target only in the World of Mathematics, which lives inside of our own heads or souls, unfortunately. In this case, they would be represented by an entity we cannot name because, even inside of our heads, we would be imagining what the eyes can see, not the World of Mathematics. As soon as you leave the World of Mathematics, you will always reach your target, therefore. The World of Mathematics can only make that set of assertions true if we talk about numbers, and, even so, only those inside of our imagination, not those we can draw somehow. It is all really abstract. In this case, it is true that the x will never reach the target, but there is no paradox, since this is all that can happen in the World of Mathematics. You get a paradox when you get conflicting conclusions from thinking of the same inferences. It suffices that we create a context, however, and there is no paradox in this case. 

That is basically my solution. 

Dear Dr. Marcia Ricci Pinheiro, thank you very much.

Might I ask three questions?

1, would you please tell me the meaning of “sigmatoid” in your post “the sigmatoid half when acquiring a non-mathematical sense and the sigmatoid half when acquiring a mathematical sense”, is it “mysterious”?

2, how and where do these two “sigmatoid halves” come from?

3, what are the relationships between these two “sigmatoid halves”?

Yours sincerely, 

Geng

Ok. Suppose let me agree that Q has infinite more elements than N. Then, what is the result for infinity + infinity. Before counting Q, you have to count Z. What is the result for Z then? Do you know the counting process is an iteration process. Counting {1,2,3,4} as {1}, {1}U{2}={1,2}, {1,2}U{3}={1,2,3} and {1,2,3}U{4}={1,2,3,4}. So, it has 4 steps. You can't immediately get 4 elements. It requires certain procedures. Because of foundations of mathematics, which you call it as foundationless mathematics. So, in order to count your so called "original" Q(i am not aware there is a "duplicate" Q), I have to subdivide them into more unique subsets.

Now forget about bijection. Countable says whether one is able to count the set. "Count able"= able to count. Mathematicians are able to count Q. That means they can arrange each element of Q in a sequence. So, Q is countable.

Dear Dr. Panchatcharam Mariappan, thank you.

You touch the root------ “what is the result for infinity + infinity”. In fact, the real trouble is: no one in the world now knows whether it is “infinite + infinite” or “potential infinite + potential infinite” or “actual infinite + actual infinite” or “potential infinite + actual infinite” or “infinite + actual infinite” or “potential infinite + infinite”?

Yours sincerely, 

Geng

http://tutorial.math.lamar.edu/Classes/CalcI/TypesOfInfinity.aspx

 Dear Dr. Panchatcharam Mariappan, thank you.

Anyone knows the “infinite” we are talking about and treating with in analysis and set theory is “infinite” or “potential infinite” or “actual infinite” or mixing ups or shiftings as one wishes…? Paradoxes come to being whenever it is missing ups or shiftings as one wishes!

Yours sincerely, 

Geng

Dear Geng

Please read this webpage:

http://www.math.vanderbilt.edu/~schectex/courses/thereals/potential.html

http://www.mlahanas.de/Greeks/Infinite.htm

and another research gate topic

https://www.researchgate.net/post/Does_actual_infinity_exist/2

A few contents from this website here.

We may see three airplanes or three apples in the physical world, but the abstract notion of "3" does not exist in the physical world -- it only exists in our minds. The notion of "3" is simple enough, and is an abstraction of enough concrete objects, that there is little chance of our disagreeing on the notion. Our conversations seem to suggest that the "3" in my head is very much like the "3" in your head (though we will never be 100% certain of that). But more complicated notions such as infinity, less grounded in physical reality, are harder to explain; it is harder to be sure that we are successfully conveying a concept from the inside of one head to the inside of another.

Formalism and its consequences were controversial at first. One of the more visible battle lines was between the group now known as classicists (who believe that mathematics is a collection of statements) and constructivists (who believe that mathematics is a collection of constructions or procedures). The overwhelming majority of mathematicians today are classicists, but this is merely a matter of personal preference (like one's favorite color), not a matter of someone being right or wrong. Nearly any mathematician today who understands both sides of the issue agrees that both sides make perfectly good sense. (On the other hand, many classicists today are entirely unfamiliar with the constructivist viewpoint.)

A striking example is the Axiom of Choice. This axiom, acceptable to classicists but not to constructivists, is a nonconstructive assertion of the "existence" of certain sets or functions. The use of the word "exist" is merely a grammatical convenience here; mathematicians and nonmathematicians do not mean quite the same thing by this word. Unfortunately, we mathematicians don't have a better word; to be more precise we would have to replace this one word with entire paragraphs. If we assume the Axiom of Choice, we are not really stating that we believe in the physical "existence" of those sets or functions. Rather, we are stating that (at least for the moment) we will agree to the convention that we are permitted write proofs in a style as though those sets or functions exist.

Whether those sets or functions "really" exist is actually not important, so long as they do not give rise to contradictions.

Dear Dr. Panchatcharam Mariappan, thank you very much!

I agree with your ideas. Studying the defects disclosed by the Harmonic Series Paradox during pass 40 years, I studied human cognizing mechanisms, the definition of human science, the differences among human cognizing fruits-------knowledges, religions, cultures, sciences,… Now I find myself in a huge working field on the way of dispelling the “2500 years old huge black cloud of infinite related paradoxes over mathematics sky”.

I feel lucky and happy to meet you here in RG and exchange our frank view points each other.

Thank you again dear Dr. Panchatcharam Mariappan.

Yours sincerely,

Geng

Dear Geng.

Thank you. I came to know about some of other infinity concepts from history because of this topic. Especially, one of the Indian mathematical method to count infinity. Here it gives a similar concept to your idea. Refer wikipedia. https://en.wikipedia.org/wiki/Infinity

The Indian mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:

Enumerable: lowest, intermediate, and highest

Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable

Infinite: nearly infinite, truly infinite, infinitely infinite

If I understand correctly, enumerable = countably finite, innumerable = countably infinite, where Q should be innumerably innumerable and uncountable should come under infinite.

The fact that Q (the positive rationals) is countable is contrary to intuition based on finite cardinalities. How could there be a pairing (one-one, onto) between Q and N?  But it is easily shown to exist (below). 

Imagine the rationals in Q arranged in an infinite array; m/n would appear in the mth column and the nth row.  Now move diagonally through the array, picking up rationals as you go but skipping rationals whose value has already appeared, to get the list

     1/1, 2/1, 1/2, 3/1, *, 1/3, 4/1, 3/2, 2/3, 1/4, 5/1, *, *, *, 1/5, ... etc.

Any positive rational m/n is listed (or equal in value to a previously listed rational) in the first (m+n)(m+n+1)/2 positions.  For example, 1073/297 is listed between positions 937765 and 939135 in this list.  THIS IS YOUR ONE-ONE, ONTO PAIRING.

I believe this pairing is due to Cantor c.1990.  All thinking about infinity before that was sort of touchy-feely and misled by finite intuition, like some of the discussion above.  Cantor made it all a logically-sound part of pure mathematics, and others since have developed the details of transfinite arithmetic. These ideas played a major role in the 'new mathematics' of elementary schools' curricula (ie. the emphasis on pairings to identify sets of equal cardinality, and very visual set theoretic definitions of + and *).  Most basic set theory texts provide this pairing and then go on to develop transfinite arithmetic. 

Dear Dr. Panchatcharam Mariappan, thank you very much!

I think of one problem: how to make sure those Enumerable, Innumerable and Infinite not to be mixed up and shifted as one wishes?

Such “not to be mixed up and shifted as one wishes” problems really have been worrying me for a long time but they should be solved of cause.

Yours sincerely,

Geng

Dear Dr. William Richard Stark, thank you.

It is true that we really can see the very diagonal proof in many books.

But is there anything wrong in my above post of “The elements of tiny portion of rational numbers from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …)  map and use up (bijective) all the numbers in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so infinite rational numbers (at least 2,3,4,5,6,…n,…) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set” and why? It is nothing to do with finite at all.

Looking forward to your frank view points, thank you.

Best Regards,

Dear Dr. Geng Ouyang,

In between 0 and 1 there are infinitely many rational numbers, similarly, between 1 and 2 and so on, implying that there are infinite rational numbers for every integer. Therefore, it cab be said that rational number set is bigger than the set of natural number .

 Dear Dr. Afaq Ahmad, thank you.

Mapping is really an important and effective idea and tool to cognize the number of elements in set theory. But no one knows whether or not we need to observe any laws for mapping---------this is one of the trouble-makers in our present classical set theory system.

Yours sincerely,

Geng

Dear Dr. Geng Ouyang, in the definition of infinite set " A set D is infinite if D is equipotent ( there is a bijection between the sets) to a proper subset of itself. So using your idea we face the same question with each infinite set. As I mentioned before read equipotent of sets and cardinal numbers. 

Dear Dr. Abdalla Ahmad Tallafha, thank you.

Cantor really opened up a new field in the infinite related mathematics and contributed a lot in set theory. But, considering some defects disclosed, something should be improved.

Yours sincerely,

Geng

Dear Geng Ouyang: ·

I believe you are saying the following:

"if B is a proper subset of A and if there is a one-one pairing between the elements of sets B and C, THEN there are more objects in A than in C, and so |A|=|C| must be false"

This reasoning is obviously correct if A is finite; but what if A is not finite?  

We intend A to be Q (Q is the set of positive rationals) and C could be any set, so replace both A and C by Q.

This substitution gives:

"if B is a proper subset of Q, and if there is a one-one pairing between B and Q THEN there are more objects in Q than in Q, and so |Q|=|Q| must be false"

But |Q|=|Q| is obviously true, so your original argument is not logical (except when applied to finite sets).

I think the answer of Professor Joseph Gielen [which is a short version of my answer] should have convinced you.

All of us who have taught set theory, foundations of mathematics or any other course which involves serious work with infinite sets, must first lead our students to abandon intuition based on the finite.

Dear Dr. Miguel Ángel Montes, thank you for the very good questions.

1, I am seriously on the new bijection relationships of the two cases in my post and state that the set of Rational numbers has more elements than the set of Natural numbers.

2, the criterion of one—one bijection by Cantor is really great and scientific. But the foundation of mapping theory has serious defects---------this is also one of the trouble-makers in our present classical set theory system.

 Best regards,

Geng

Dear Dr. William Richard Stark, thank you.

Re: I think the answer of Professor Joseph Gielen [which is a short version of my answer] should have convinced you.

According to the definition, the original Even Number Set and Odd Number Set is each half of the Natural Number Set (otherwise it can not be called Even Number Set and Odd Number Set). People have been using two deformed Natural Number Set of (1x2, 2x2, 3x2,...,nx2,...) and (1x2+1, 2x2+1, 3x2+1,...,nx2+1,...)  and just used the components inside even numbers and odd numbers (natural number components of “1, 2, 3, 4, 5, 6, …, n …”, but not the whole thing of “2,4,6,8…2n” and “1,3,5,7,9…nx2+1”) to map natural numbers, of cause they map very well because it is N ----> N itself but not 2n ----> N and nx2+1----> N. 

1, the bijection relationships of the case we discuss above are in the infinite field, nothing to do with finite.

2, what are the mappings to natural numbers: the even numbers and odd numbers or the components of even numbers and odd numbers?!

I think Dr. Joseph Gielen and I have totally different way of mapping for Natural Number Set and Even Number Set. So, his idea has nothing to do with my conclusion. 

Yours sincerely,

Geng

I was saying that the  logic  of your concern came from our experiences with finite sets, not that you were using finite sets.  Clearly, "half a set A is smaller than the whole set A" is a valid statement for a finite set A, but not for an infinite A.

give me please the definition of   A  biger than  B ( where  A,B- infinite)

in mathematics every thing is depends on definition

the countability it does not mean anything except countability as a definition

Dear Dr. Miguel Ángel Montes, Dr. William Richard Stark and Dr. Ramadan Sabra, thank you.

@ Miguel: would you please give your frank points of view to following two questions?

(1), are you sure there are really any ”principals, laws or criteria” for what and how to map elements between two sets in present mathematics, where can we find them and what are they?

(2), Is there anything wrong in my post of “The elements of tiny portion of rational numbers from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …)  map and use up (bijective) all the numbers in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so infinite rational numbers (at least 2,3,4,5,6,…n,…) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set ------- Rational Number Set has infinite more elements than Natural Number Set”?

@ William: I sincerely hope you can give your frank points of view to following two questions?

(1), Infinite Rational Number Set and Infinite Natural Number Set in my post are really Infinite sets, what are finite logic and infinite logic?

(2), would you please point out frankly and exactly what kind of mistakes there are in my post of “The elements of tiny portion of rational numbers from Infinite Rational Number 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 Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set ------- Rational Number Set has infinite more elements than Natural Number Set”?

@ Ramadan: we may not worry about “bigger or smaller”, “countable or uncountable”, “cardinality” first.

Would you please point out frankly and exactly what kind of mistakes there are in my post of “The elements of tiny portion of rational numbers from Infinite Rational Number 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 Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set ------- Rational Number Set has infinite more elements than Natural Number Set”?

My heartfelt thanks to you all and looking forward to your answers.

Yours sincerely,

Geng

you are right the natural numbers N satisfies  this property and it is countable and satisfies other properties

my question is : what is this property called? ( definition)

Dear Dr. Ramadan Sabra, thank you.

Re: you are right the natural numbers N satisfies  this property.

What property do you mean? Do you mean Rational Number Set has infinite more elements than Natural Number Set?

Yours sincerely,

Geng

Dear Geng Ouyang

No

  I mean that there are many subsets of N   (  A,B,C,...) and  many bijections 

N to  Q

A to Q

B  to Q

Remark:  there are  bijections 

N  to   N

N  to  A

N  to   B

and  N   has numbers more than  each of  A,B,C ,.....

-  so I think  there is a deference between the theory of finite and infinite sets

- and we do not understand ( or catch) the infinity !!!!!!!!

--there is no definition of infinity!!!!!!!!

-- what is the infinity and what is the infinity plus 1  ,  who is biger?

Regards 

Hello Ramadan,

there are answers to all these points in the theories of  ordinals (Cantor, Gödel, ...).

The only question of interest in the continuum hypothesis.

The other points on the definition of the infinite are not in the research's scope anymore.

Best Regards,

 Gabriel

Dear Geng Ouyang

could you show  the contraduction in your problem please

Regards

Dear Dr. Ramadan Sabra, thank you.

1, In present infinite related mathematics theory system, on the one hand, we can prove Rational Number Set has infinite more elements than Natural Number Set------no bijective relationship between them at all; on the other hand, we really can see the very diagonal proof in many books by Cantor. What and how do we choose, why?

2, According to the definition, the Natural Number Set can be divided into Even Number Set and Odd Number Set (otherwise it can not be called Natural Number Set, Even Number Set and Odd Number Set). People have been using two deformed Natural Number Set of (1x2, 2x2, 3x2,...,nx2,...) and (1x2+1, 2x2+1, 3x2+1,...,nx2+1,...)  and just used the components of even numbers and odd numbers (natural numbers, but not the whole thing of “2,4,6,8…2n” and “1,3,5,7,9…nx2+1”) to map natural numbers, of cause they map very well because it is N ----> N itself but not 2n ----> N and nx2+1----> N. 

What are the mappings to natural numbers: the even numbers and odd numbers or the components of even numbers and odd numbers?!

Mapping is really an important and effective idea and tool to cognize the number of elements in set theory. But no one knows whether or not we need to observe any laws for mapping---------this is one of the trouble-makers in our present classical set theory system.

Yours sincerely,

Geng

Thank you Gabriel