1, How many definitions of “infinite” with different natures in science can we have as ideas of human mind?

2, How many definitions of “infinite” with different natures in science can we have as something we can find in the real world?

3，How many definitions of “infinite” with different natures can we have as co-product of human mind and objectivity in science?

4, in front of so many different definitions with different natures for the concept of “infinite” in human science, how can we distinguish them theoretically and practically?

There are many arguments on “infinite-finite” in RG, but we can see they actually are the debates between “potential infinite” and “actual infinite”. Without knowing what “finite-infinite” and “potential infinite-actual infinite” are philosophically and mathematically, these debates will last forever.

Hi, Miguel

Thank you for your question. I have reedited the 2.

As you can see, there are many topics discussing “infinitude” in RG, I found following 2 arguing problems:

1, we can have different definitions of “infinitude” with different natures in science; is it true or how many?

2, “infinitude things” are just products of human mind or we can find “infinitude things” in the real world?

So, one of the purposes of my question is to discuss how much is “infinitude” in science relating to human mind and objectivity.

Cardinal number of the set of natural numbers: is it objective? Of course it is.

Cardinal number of real numbers: the same.

´Objective´ =  exist in the real world? How do you define real world?

Real world contains just concrete (in principle localizable in space and time) objects and processes. Abstract objects arise from abstracting activities but then they are objective. (But not being concrete they have no existence (see Bolzano 1837).)

I have added my answeer, who has annihilated it???

Geng Ouyang, I am told that yóu reacted to my answer. Can I know your reaction?

Dear Mr. Pavel Materna,

Thank you for the good question, “objectivity” and “subjectivity”, it is very philosophy.

In my opinion, we have “theoretical philosophy—applied philosophy (such as philosophy of mathematics)” as our cognized fruits in science.

Many people have been arguing the natures and the demarcation of “objectivity” and “subjectivity” in both “theoretical philosophy” and “applied philosophy”. I guess this argument will last forever.

I am now working in the period of philosophy of mathematics, so I accept present ideas of “objectivity” and “subjectivity” in “applied philosophy” thought it may be not that perfect: objectivity = real world = the world which human (human mind) cognize…. Of cause human is also part the of real world if we want to cognize ourselves; but for mathematicians, human (human mind) and the world being cognized by human are two different things.

I appreciate your frank opinions very much.

Regards,

Geng Ouyang

Paul Ricœur parle de « constitution ontologique instable » par laquelle l’homme est à la fois plus grand et plus petit que lui-même, la faillibilité se trouvant de ce fait dans la disproportion entre finitude et infinitude. — (Jacques Sys, Les imaginaires christologiques, 2000) 

Paul Ricœur speaks about " unstable ontological constitution " by which the man is bigger and smaller at the same time than himself, the fallibility being of this fact in the disproportion between finiteness and indefiniteness. - (Jacques Sys, imagination christologiques, on 2000)

examples of groupe that applicate in nature or transcomunication their mind of anderstanding the concept of infinitude 

http://www.verslinfinitude.fr/

http://www.infinitude.asso.fr/

Dear Miguel，

What I say “human (human mind) and the world being cognized by human are two different things” means objective and subjective are two different things to mathematicians and a very important thing for mathematicians is to make sure their work the proportion between objective and subjective---------the very fundamental work for infinite theory system in our mathematics. You can see many of my posts in RG mentioning the proportion between objective and subjective.

Thank you!

maybe this text about infinitude in the time for Aristotle can be usefull for your reflexions?

https://es.scribd.com/doc/170131168/ARISTOTELES-Y-LA-INFINITUD-EXTENSIVA-DEL-TIEMPO

Hi Cecile,

According to my studies on infinite relating paradoxes (I have uploaded 2 papers about studies on infinite relating paradoxes on the RG), in front of the suspended Zeno’s Paradox family, our human’s wisdom and science have nothing improving at all: the foundation of present infinite theory system is exactly the same thing as that of Zeno’s time. So, more exactly, to present infinite theory system, we are living in Zeno’s time 2500 years ago.

Thank you!

Dear Miguel，

To say a truth, you are the most professional researcher in the field of infinite I have ever met in RG, thank you. Why I say 'to present infinite theory system, we are living in Zeno’s time 2500 years ago' in my last post based on following 2 points:

1, theoretically, in present infinite theory system even after many researchers and Cantor's work, the nature of “infinite” is still very confusing: can we have only one definition for the concept of “infinite” in our science (including mathematics) or many; on the other hand, can we have different degree of “infinite”------ more “infinite”, more and more “infinite”,…; less “infinite”, less and less “infinite”, … ;…?

2, practically, Cantor and many researchers cared less about the nature and the practical “behavior” of infinitesimal, any researchers’ work (including limit theory and Cantor's work) did nothing to solve the defects discovered 2500 years ago by Zeno’s Paradox family of “Achilles--Turtle Race” in present classical infinite theory system.

We have come a long way from Zeno's times. Modern set theory has advanced our thinking to finiteness and therefore infiniteness. Perhaps it is the other way around. We are still referencing our philosophical underpinnings to define infinity. But Miguel is right . Today infiniteness refers more to objects. Infiniteness is finite undefined.

Being finite beings our practical use of infinity is very limited to intellectual  "lava"

interestingly, I thought fractal geometry attempts to provide some implicit answers to definition of infinity. Mandlbrot tried to bring in the needed objectivity to infiniteness in a limited way.........

Dear Mr. Raveendran Iyengar,

You are right, “We have come a long way from Zeno's times...”; but the problem is theoretically or practically, we did nothing to solve the defects discovered 2500 years ago by Zeno’s Paradox family of “Achilles--Turtle Race”, it is why I say that we are living in Zeno’s time 2500 years ago'.

Talking about Cantor's view of infinity, I am a little sorry to say that he cared little for infinitesimal. Does infinitesimal belong to infinite and is infinitesimal anything to do with cardinal numbers or ordinal numbers?

So, that is why I call Cantor's view of infinity “half infinite”; I know many people nowadays will not agree with me, but what I have said is true.

Geng Ouyang, You have highlighted one of the most interesting points in set theory, namely, the place of infinitesimal. While infinitude is object, the reference to infinitesimal has always been ordinal. At least that is what my understanding has been. Otherwise, calculus will have problems. But what cantor was trying to do is to point a big picture. I for one will agree with you that his view is "half infinite"

Dear Mr. Raveendran Iyengar,

Thank you!

The suspended Zeno’s infinitesimal related paradox family of “Achilles--Turtle Race” decides that in infinitesimal area, there are really a lot of things waiting to be done.

Regards!

Geng

Mmmmmmmmmmmmmm.......

Miguel, a very nice thought. . 1 / 00 ( read as  one divided by infinity ) . If one thinks of both infinity and infinitesimal together , it is strange.It is like trying to visioning the point of singularity where everything stretches to infinity.  I would love to hear any articulation of the relationship .........

Hi, Miguel,

I guess the word “infinity” you use is infinite but not “infinite big opposite of infinite small” (I was often misunderstood before, so I use “infinitude”). So, I think infinitesimals are infinite related small mathematical number forms.

A set A is infinite if and only if there is a function f : A ---> A that is injective but not surjective. This definition is very objective, and has nothing to do with any tricky construction of the human mind. For example  f: N ---> N given by f(n) = n+1. If you do not agree with this, then I must ask you what is the last natural number...  The idea that the set of natural numbers is potentially infinite because we can think of it only up to a very big number, but the interval [0, 1] is actually infinite, because it lies under our eyes with its whole infinity of points, is quite fallacious. In fact, the sequence { 1/n | n > 0 } lies in the interval [0,1] so is actually infinite, and is in bijection with the set of natural numbers - so the set of natural numbers is also actually infinite. In conclusion, we do not need the distinction between actual and potential infinity. I find more important the distinction between infinite sets of different cardinalities, like countable infinite sets (like N) and continuums (like [0,1]) , respectively sets of bigger cardinality.

Hi, Miguel,

Since the concept of infinite was created basing on objectivity, people began to cognize “infinite things” in mathematics from quantitative angle; so, “infinite small quantitative things------ infinitesimals” and “infinite big quantitative things------infinity” came into mathematics. So, I call it ‘Infinite related small mathematical number forms'.

But because of the infinite related fundamental defects in science (including mathematics) we human actually can not understand the natures of “infinity” and “infinitesimal”. People took them as numbers in all kinds of calculations but never acknowledge them as numbers, what you say is right: “that leads you to a bunch of questions”. This is one of the very reasons for me began my studies 40 years ago.

According to my studies, zero is not “small mathematical number, which has a relationship with infinity” as you say; this is one of the mistakes in present traditional infinite theory system, zero is absolutely nothing.

I don’t quite catch the meaning of your question, is it the relationship between “infinity” and “infinitesimal”: “infinite small quantitative things------ infinitesimals” and “infinite big quantitative things------ infinity”?

@ Miguel Angel Montes

. ¿Can you try a definition of infinity that does not concern sets?

Well, infinity is a property of sets. Better said, cardinality is a property of sets, and infinity is a property shared by some cardinalities... Yes, it is hard to define infinity without using sets, but why should we do this? 

About analytic and mechanical notions concerning infinity, like "infinitely small movement", and so on: This is a quite different question from the first one, about potential and actual infinity. Here there is much more structure involved. Archimede's axiom tells that for every real number, there is a natural number that is bigger than it. Happilly, there are ordered fields which are non-archimedean. There you have field elements r which are bigger than any natural number. Than 1 / r will be positive but smaller than all 1/n , so this is an "infinitely small quantity" (like in non-standard analysis). As you see, infinitely small elements are a notion which cannot be befined without additional structures (ordered fields, order compatible with field structure, negation of Archimede's axiom, and so on) and not only a discussion about cardinalities, like the other point.

However, I agree with your arguments concerning psychology and philosophy, or the way children can learn about infinity. Fine psychological mechanisms maybe continue to make the difference between actual and potential infinity even by professional mathematicians. The most intense use of potential infinity appears, as I believe, when cosmologists try to think about an infinite universe and when analytical mechanicians try to think about an infinitely small movement...

Pupils understand immediately that there is not something like a biggest natural number. So they understand that the set N is infinite. Also, they understand that if n becomes bigger and bigger, 1/n tends to become 0, without reaching 0. So, the next step is to explain them that "n becomes bigger and bigger" is the same thing as saying "n tends to infinity, without reaching it". Children have a sense of continuity, and then we tell them that "infinity" is introduced for the sake of completion. I know that pupils of fifth class or sixth class will understand this. 

Infinity understanding is better served along with a better understanding of truth and belief.  Uncountable set P(N)  relies on an unique understanding of both belief and truth. 

Taken in isolation , envisioning infinity is well neigh impossible.....

Dear Raveendran,

It is very good to discuss how to teach kids some difficult scientific concepts, I remembered once my son asked me “how do I come to this world?” when he was 5, I tried many ways….

I totally agree with Raveendran, it is very important that “Infinity understanding is better served along with a better understanding of truth and belief”.

In fact, so many people have offered ideas and methods to “analysis, solve” Zeno’s Paradox (it is easy) since antiquity, but the problem is in front of the practical case they are unable to do anything. When we ask: how and when dose Achilles catch up with the Turtle? No one in the world so far can offer rigorous answer theoretically and practically. This is the point.

So, In present traditional “potential infinite--actual infinite” based infinite theory system, in front of non-standard analysis, standard analysis with limit theory is not rigorous and in front of Zeno’s Paradox of “Achilles--Turtle Race”, standard analysis and non-standard analysis are impossible to be rigorous.

Very true Geng. Ouyang. Our perception of  Infinity is " inside out" view where as any comprehensive theory will have to take an "outside in " view. I am not sure we have the capabilities to do that . 

But one can never know........

Dear Raveendran,

I think this is a very open forum, can you tell us your idea about “But one can never know what?”

What I meant was right at present the current thinking on infinity is , we are trying to understand something from being the part of that. 

We may perhaps need a view that looks at from outside. Our thinking faculties are limited therefore. But one can never know, it may soon dawn upon some one , like copurnicus or galilio or Newton or Einstien........ That s what I meant...

The infinite related area is a huge one, but we can do want we can in the rest of our life.

@ Ellis，

So, you use mental models to illustrate all kinds of human knowledge such as the mental models of mathematics, the mental models of English, the mental models of religious…?

I think this idea is new to many people. Is mental model “a state of the mind”, a “belief”,…?

Regards!

Geng

Migue,

I guess you are talking about “intuition and insight”.

But do you care for “personal intuition and personal insight” or care for “scientific intuition and scientific insight”?

Migue,

Many people agree that we scientists work on the bridge between “objectivity” and “subjectivity”, either too much “objectivity” or too much “subjectivity” is not "Scientifically".

So, in my opinion, “personal intuition and personal insight” is very possible to be more “subjectivity” (it even can be personal belief or religious) while “scientific intuition and scientific insight” is well balanced.

Of cause it is sometimes difficult to distinguish between “personal” and “scientific” because at that time “looks very personal: a person has scientific intuition and insight”. May be that is why  ‘many great genius said, in his time, things that weren't considered science’.

There are many debates on “infinite-finite”, but we can see they actually are the arguments between “potential infinite” and “actual infinite”. Without knowing what “potential infinite” and “actual infinite” are philosophically and mathematically, these arguments and debates is surely endless.

“In terms of values and in terms of repetition” is the idea for two kinds of infinite since immemorial.

But the problem is different people have different understandings and interpretations for these two infinites and their relationship (different version?) since immemorial------it is very difficult to know what is what in front of the practical infinite related cases, and these are the real reasons why those infinite related paradoxes have been produced and unsolvable at least since Zero’ time 2500 years ago.

I come back because the question changed a little bit, coming to the paradigm "potential infinity vs. actual infinity". People don't stop to debate about infinity because infinity is hard to be imagined, even if you are a mathematician and you have thought about infinite sets along your whole life. Potential infinity is somehow easier to imagine as actual infinity, and this is a psychological phenomenon. I guess that it happens because imagination itself is more a potentiality as an actuality, being always uncomplete. This (felt!) difference in the difficulty to imagine them leads to debates. However, the potential infinity could not exist without the actual infinity - or better said, the potential infinity is a psychological manifestation of the actual infinity - and so we would do better to accept both of them with equal effort.

Dear Mihai and Ellis,

In fact, “potential infinite” and “actual infinite” do not only exist as “imagination and psychological phenomenon”, there are many PRACTICAL EXAMPLES IN MATHEMATICS.

Let’s see following divergent proof of Harmonic Series which can still be found in many current higher mathematical books written in all kinds of languages:

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）

The unavoidable practical trouble is how many items in infinite Harmonic Series can be added up by “brackets-placing rule" to produce infinite numbers each bigger than 1/2: “Potential infinite” or “actual infinite” decides whether or not we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from Harmonic Series by “brackets-placing rule" to change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity.

Sincerely yours, Geng

Dear Mihai and Ellis,

Let’s think about a very primitive and general question from the concept definition theory lever: can we have two or three or more different definitions with different natures on one concept in our science (such as “infinite”: in terms of repetition or in terms of values or in terms of mathematics or in terms of physics or in terms of psychology or in terms of linguistics or in terms of kids or in terms of adults or…)? If YES, how can we distinguish them in practices to avoid suspended infinite related paradoxes? If NO, …!

Actually, we human have been troubled by this question at lease since Zeno’s time 2500 years ago and people never stop debating on “potential infinite and actual infinite” ever since.

Sincerely yours, Geng

Dear Geng,

your example is still very anthropomorphic. The object called "infinite series" is very linked with human significancy (a text, readable from left to right, which as a symbol has a significance, etc...) and two slidly different interpretations of this text (called infinite series) can be done, such that the old problem of potential vs. actual infinity seems to show back. The object is maybe too complicated. One has the same discussion by looking at the sequence (1/n) for n > 0. The POTENTIAL view: one looks to the point 1/n how it slowly approaches 0 when n goes to infinity. The velocity is smaller and smaller and the steps become shorter and shorter. It does never touch 0, but it approaches 0 at infinity. The ACTUAL view: all the elements of the infinite sequence are already there! They are so that for any neighborhood of 0 there is an N depending of the neighborhood, such that for all n > N, all elements 1/n belong to the given neighborhood. As you know, the potential view is usual for kids to understand what does convergence mean. On the other hand, the actual view, which is more clear and has more rigor, is the way in which  the notion of convergence is defined and the way one works with those things, in the moment that they have been better understood. There are only two ways to see the same think, and they differ only psychologically. Of course, both of them do exist - as any other psychological perception also exists. That was not the point of non-existence here.

Dear Mihai,

Could you please tell me your frank opinion to following two questions:

1, can we say “small infinite” and “big infinite”?

2，can we really use that “brackets-placing rule" to produce infinite numbers each bigger than 1/2 from Harmonic Series?

Sincerely yours, Geng

Dear Ms. Ellis D. Cooper,

Can you share your idea with us why “for sure, there are no infinities in the physical universe”? Is physical universe that much different from mathematical universe, what is the relationship of physics and mathematics you think?

Would you please tell me how you think of the “potential infinite--actual infinite” related Harmonic Series Paradox in above post, it is a practical problem.

Sincerely yours, Geng

Dear Ms. Ellis D. Cooper,

It is believed that mathematical universe has a lot to do with objective world except mental models amongst less than half a million living human beings because it is one of human sciences (a balance and harmonious compound of "objective and subjective" worlds) with verbal, diagrammatic… mathematical formal languages.

Many people believe that physicists only take mathematics as a tool and work with “applied mathematics”, but the foundation of “applied mathematics” is “theoretical mathematics” which has very close relationship with infinities.

If both mathematics and physics are not only the imaginations or mental models of us human beings but have much to do with objective world as well, it should have very close relationship with infinities.

Sincerely yours, Geng

I agree with Geng. To add a bit more to that, our understanding of physics and mathematics is a perspective of inside out view. Imagine, looking at  a city 's skyline from our window. Our vision is only partial even though we might get it clearly.  

One cannot argue that what is not visible from our window does not exist. Same is true for mathematics and physics. The underpinning of theoretical sciences is human ingenuity. Whilst the empirical evidence may take time to produce, it does not in anyway take away the credibility of original thinking.

In short Infinity , truth and belief are closely linked......... In my opinion.

Dear Raveendran,

As most people agree that human science is a kind of human’s property (some other creatures in universe may also “enjoy” this property) composed by human’s mental work (subjective world) and universe (objective world) ------- no human science without human. So, the unbalanced things of subjective and objective compounds of our knowledge are not scientific. Sometimes the subjective factor or objective factor is overstated, each causes confusion.

Human science is human’s, so we must present all the things in our science with our human’s ways. Our science history proves that this is not easy even very hard, the infinite related things in our science are typical examples-------- people have been trying very hard for at least 2500 years so far but the arguments about “finite-finite”, “potential infinite” and “actual infinite” aroused by the suspended infinite related paradoxes are still on and on endless. Something should be done in this field to improve our science.

Regards, Geng

It is possible that peoples of different linguistic backgrounds conceptualise infinity differently due to peculiarities of their languages. I came across an interesting article about this problem on RG . The paper was written by Dong-Joong Kim, Joan Ferrini-Mundy and Anna Sfard; it is titled: "How does language impact the learning of mathematics? Comparison of English and Korean speaking university students’ discourses on infinity".

The article claims that "It was found that in spite of the comparable levels of mathematical performance, there was, indeed, a visible dissimilarity between mathematical discourses on infinity of Korean- and English speaking students. In general, whereas no group could pride itself on a well-developed mathematical discourse on infinity, the mathematical discourse of the English speakers, just like their colloquial discourse, was predominantly processual, whereas the Korean speaking students’ talk on infinity was more structural and, in an admittedly superficial way, closer to the formal mathematical discourse." 

This could explain the inconclusive results of many interesting discussions initiated by Geng. You can find the article on Anna Sfard ResearchGate site. 

Dear Mr., Untangling Math, thank your for your frank opinions.

But I think there may be 2 other possible situations too:

1, mistaken understanding or mistaken translation,

2, some new ideas on infinite,

thank you again!

Yours,

Geng.

Hi, 

RE: "But I think there may be 2 other possible situations too: 1, mistaken understanding or mistaken translation,"

True, wrong translation is always a possibility, but the authors of the article seem to include native speakers of Korean. 

RE: "2, some new ideas on infinite," 

There is nothing wrong with redefining mathematical concepts. The problem is that it is not clear whether you want to redefine the concept of infinity within the context of mathematics because you seem to jump between common understanding and mathematical understanding of it. The way I see it, you criticise mathematical definitions for not always relating well to our everyday experiences and notions. 

Regards, UM

Dear Mr. Untangling Math，

The suspended paradoxes drive us logically to solve the defects in the foundation of our science and mathematics, including redefining some mistaken concepts.

Best regards,

Geng

“Potential infinite” and “actual infinite” have been two important concepts in our present classical infinite related foundation of science (mathematics) theory system at lest since Zeno’s time more than 2500 years ago. The defects in this field have produced “the 2500 years old black cloud of infinite related paradoxes over mathematics sky.”

1, When paying equal attention on “potential infinite” and “actual infinite”, we meet the uncompromised logical contradiction and the typical case is the “strict mathematical proven” ancient Zeno’s Paradoxes and modern Paradox of Harmonic Series.

2, When paying more attention on “potential infinite” and neglect “actual infinite”, we meet all the ideas and theories of “all infinite things are the same endless without any inborn and numerical natures” and the typical cases are all the ideas and theories of the countable infinite sets.

3, When paying more attention on “actual infinite” and neglect “potential infinite”, we meet all the ideas and theories of “all infinite things are not the same limitless and endless, they have inborn and numerical natures” and the typical cases are all the ideas and theories of power set and uncountable infinite sets (infinite Set R has more elements than infinite Set N): more infinite, more more infinite, more more more infinite,…; higher infinite, higher higher infinite, higher higher higher infinite,…; bigger infinite, bigger bigger infinite, bigger bigger bigger infinite,…; supper infinite, supper supper infinite, supper supper supper infinite,…; Continuum Hypothesis,…

Dear colleagues, anyone knows how to solve following problems:

1, 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 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?!

2,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 and they are not bijective at all.

What decides the same cardinality of two sets?

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.

Regards,

Geng

Hi Geng, 

What do you mean by "map and use up all the numbers (elements) in Natural Number Set"?

The concept of ALL makes no sense in the context of infinite sets. ALL  is associated with ordinary sets because they are finite. You cannot import the idea of ALL from finite sets to infinite sets.

Best, Wes

Dear Wes, thank you.

By "map and use up all" I mean bijective.

Best,

Geng

Hi Geng, 

Regarding "By "map and use up all" I mean bijective."

If you really meant that, you would not pose the question "What decides the same cardinality of two sets?" 

The term "bijection" is a technical concepts. It is clearly defined and there is no notion of ALL in it. It is what it is, and you cannot argue with it. It looks to me that you want the word mean more than it actually does.

Best, Wes 

 Dear Professor Wes Raykowski, thank you very much for your kind and friendly suggestions.

By“It looks to me that you want the word mean more than bijection actually does”in your post, would you please suggest some other terms for “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 all the elements (bijective) in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …)” in present classical infinite related science system.

Thank you in advance.

Best,

Geng

Hi Geng, 

As I mentioned before I am not a professor, just a friendly neighborhood researcher. 

What you are describing is not based on the mathematical notion of infinity which is defined in a strictly technical way. This definition was made to satisfy logic, not human psychological needs. To my mind, your question is illogical in this context so there is no answer to it. 

Sorry Geng, the world of mathematics is very cruel and unforgiving. 

Best, Wes

 Dear Wes, thank you.

1, if as you put it: the mathematical notion of infinite is defined in a strictly technical way, why we human have hard debates on “what infinite is?” and “what potential infinite and actual infinite are?” and so many members of suspended infinite related paradox families have been challenging us at least since Zeon’s time 2500 years ago?

2, you are right that the world of mathematics is very cruel and unforgiving; but the world of mathematics is constructed, maintained and developed by human, the contents produced by the defects been disclosed should be solved by human themselves------this is the metabolism natural law. 

Best,

Geng

Five of my published papers have been up loaded onto RG to answer such questions:

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