Thank you dear Miguel, you are always so friendly and helpful.

1, I think in mathematics, we only accept the first and the second way in your post (by Efraim Fishbein) but not the third way to understand infinity, and do you agree?

2, actually, the first and the second way in your post (by Efraim Fishbein) to understand infinity never tell us what the distinction is between “potential infinite” and “actual infinite”. The most simple, popular and troublesome problem we often meet is: when we are facing an “infinite thing”, we just don’t know what kind of “infinite” it should be and what to do------“potential infinite” or “actual infinite”, what to do if “potential infinite” and what to do if “actual infinite”? Harmonic Series Paradox is a typical example.

Yours, Geng

Dear Miguel, thank you for your frank opinion.

1, In formal mathematics we don’t just accept “actual infinite” but also unavoidable “potential infinite”, otherwise there would not be the divergent proof of Harmonic Series.

2, If “there is a difference between what infinity is (or should be) for mathematicians and what is for non-experts”, then we mathematicians should improve our work to make the DIFFERENCE disappear.

3, “When you face a mathematical situation when infinity (the mathematical one) in involved, the way in which you face the situation, is what determines if your cognition about infinity is actual or potential.”  That is right, but the problem is no one is able to distinguish them------ Harmonic Series Paradox is a typical example.

4, Harmonic Series Paradox is a typical mathematical example but not psychological.

Best regards, Geng

1) Both of these terms are largely matters of the philosophy of mathematics and even theology (see e.g., the many works on this by theologian William L. Craig). Within physics, we do not find this distinction made much, but rather we find (as in mathematics) references to singularities.

2) Within mathematics, there isn't any one infinity. For those to whom cardinals are of minimal importance, this distinction boils down to countable vs. uncountable infinities.

3) "Actual infinite" means mostly "stop using calculus/analysis and other mathematics", in that while in mathematics it is easy enough to sum infinitely many terms, take limits at infinity, refer to both positive and negative infinity, refer to infinitely many infinities, and refer to infinite-dimensional space (i.e., not space that extends infinitely but does so "along" infinitely many "directions/dimensions"). A lot of research within physics asserts that the answer Zeno's Achilles/Tortoise " isn't correct, in that spatial or spatio-temporal distances are not infinitely divisible. Hilbert's hotel clearly can't exist in any accepted model of our universe. Yet temperatures below absolute 0 appear to be greater than the infinite number of all positive temperatures, representing perhaps an "actual infinity".

Basically, as infinity has a vastly more diverse number of contexts, instantiations, interpretations, and uses within mathematics that do not correspond (or may not correspond) to anything within physics/the "actual/real" world, the distinction is intended (usually) to reflect this.

Actual infinity was a very strange concept for ancient scientist. They were using a potential infinity because of some philosophical issues. For them potential infinity means, that you deal with numbers which could be increased/decreases. This process is without any end, but in fact you deal with finite numbers.

The first attempts to understand actual infinity were done by philosophers... and theologians. The first people who writes about a infinite set was Saint Augustine. It was a part of theological considerations. 

Recommendation: Read Charles Seife's book "Zero. The Biography of a Dangerous Idea". .It is simple, elegant and witty. It decribes the zero and its partner, infinity, through the evolution of our civilization.

Mathematically one can always define the singularity causing "infinite behaviour" so as to satisfy most of our needs. I am not comfortable with loose concepts like potential and actual infinity.

Note though that present advances in physics, chemistry and biology might give a most "pregnant" understanding of the concept of Zero, see e.g. Lawrence Krauss – and if zero is pregnant so is infinity!

Dear Miguel, Andrew, Paweł and Erkki,

Actually, you four researchers have different understandings each other on "infinite"------it is really a challenging problem.

Sincerely yours,

Geng

The importance of this is one of practicality.  In math two things that approach each other but next quite get there happens all the time.  This is the infinitely smaller scenario but in reality that does not happen so we need the differences to use for what it seems like and what reality tells us is true.  On the other side of this if two things are going away from each other at a constant speed and they have nothing in the paths to stop or change their paths they will head to infinity but can never get there.  As a matter of fact they will never be an infinite distance from one other.  So we need the concept even if we can not demonstrate the concept.  In the universe there is not way to show that there is an infinite distance but in math we can. 

I think that we run into the problem as this is a conceptual issue that only has meaning in the abstract and true infinity exists but can never be used in the practical.   

Miguel Ángel Montes's replies are predicated on acceptance of the philosophy of mathematics most commonly accepted (implicitly or explicitly) among mathematicians working in areas distant from computability theory, and are sound expositions of the matter within the context of that view.

I would note, however, that there actually are finitists who deny the existence of any actual infinities, and that their view, that only potential infinities exist, is held as a practical matter, if not a philosophical position, by anyone working in domains such as resource-bounded inference.

Nor, even within a more-or-less classical setting is Cantor's definition the only definition of infinite:  one could also define infinite as not-Kuratowski-subfinite.  A set (or object in an elementary topos) is Kuratowski finite if the free join semilattice generated by it (there is a trick that lets one define free join semilattices, unlike general free algebras, by a purely finitistic construction), and a set (object) is Kuratowski subfinite if it is a subset (subobject) of a Kuratowski finite set (object).

Dear George， you hit the point:

“So we need the concept even if we can not demonstrate the concept.”, “I think that we run into the problem as this is a conceptual issue that only has meaning in the abstract and true infinity exists but can never be used in the practical.”

Many people believe this helpless situation since Zero’ time 2500 years ago and have been trying hard to solve the problem. Looking back into our 2500 years mathematics (science) history, “potential infinite” and “actual infinite” are really there in our present traditional mathematics (science) and we can see so many people talk about what they are in all kinds of papers and books, but the problem is those researchers actually know little about them------so the suspended infinite related paradoxes have been there ever since. The infinite related mathematics (science) foundation is really a challenging field for us to pioneer and develop.

Sincerely yours, Geng

Dear David,

Thank you very much for your constructive and frank opinion.

What you say is true “that there actually are finitists who deny the existence of any actual infinities, and that their view, that only potential infinities exist, is held as a practical matter.” But in our real life, we have to face many infinite related cases in mathematics (science).

Would you please be so kind and frank as to telling me how do you think of the unavoidable practical trouble of “potential infinite--actual infinite” in above Harmonic Series Paradox?

Sincerely yours, Geng

Dear Wes,

Thank you very much for your theoretical opinions.

Looking back into our 2500 years mathematics (science) history, “potential infinite” and “actual infinite” are really there in our present traditional mathematics (science) and we can see in fact a hundred people can have a hundred version of “potential infinite” and “actual infinite”. But in our real life, we have to face many practical infinite related cases in mathematics (science).

Would you please be so kind and frank as to telling me how do you think of the unavoidable practical trouble of “potential infinite--actual infinite” in above Harmonic Series Paradox?

Sincerely yours, Geng

Dear Wes,

Thank you for your thoughtful reply.

I agree with you, the summary text from Wikipedia is widely accepted because “in terms of values and in terms of repetition” is the idea for two kinds of infinite since immemorial.

But do you notice, 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.

Your reply is really helpful and friendly, I am happy and thankful to accept your term of ‘harmonic series’. I know very well that my English is not good enough and I sometimes really doubt whether my expressions can be well understood in RG.

Sincerely yours, Geng

I am always amazed by how one can philosophize and philosophize and philosophize ... without end about such a simple word as "infinity" (the property of having no end), making the concept infinitely complicated, making the explanation of infinity infinite.

Dear Robert,

Many people really have the same feeling with you!

Looking back into our mathematics (science) history, I think for at least 2500 years people really somewhat have been “philosophized and philosophized”------it is those suspended infinite related paradoxes (such as family members of Zeno’s Paradox) in present science (mathematics) that attract and drive people working so hard on “infinite”.

But it is not true now; we are now in this thread working on the practical mathematical cases such as newly discovered Harmonic Series Paradox in above post. The infinite related mathematics (science) foundation is really a challenging field for us to pioneer and develop.

Sincerely yours, Geng

Dear Wes,

It is true that we really can see so many people talk about what “infinite”, “potential infinite”, “actual infinite.” and limit (delta-epsilon definition and operation) are in all kinds of papers and books, but the problem is those suspended infinite related paradoxes (such as family members of Zeno’s Paradox) strongly disclose the systematic defects in present traditional infinite related cognitions. In front of family members of suspended infinite related paradoxes we really have to do something practical but not going on and on “philosophized and philosophized” as Mr. Robert Laurence Baber just points out.

Wes, could you tell me frankly your idea from the cognition point of view: can we have two different definitions with different natures on the concept of “infinite”.

Thank you Wes!

Sincerely yours, Geng

Re the question of Wes Raikowski:

The word "infinity is composed of three parts: the prefix in-, meaning not, the stem fin meaning end and the suffix -ity meaning property. The word itself means, therefore, the property of having no end -- no more, no less. The thing to which one refers, i.e. the "what" in your question, must come from the context, i.e. from the rest of the text in which the word is used.

Perhaps by not specifying the "what" to which you refer one opens up an endless (infinite) range of things to philosophize about.

Dear Miguel,

I shudder at the thought that mathematicians do not use a "linguistically" defined language. Especially in mathematics it is critically important that one chooses and uses language impeccably clearly.

The open interval (0, 1) has no end, i.e. is infinite.

Regarding your comment "Considering infinity as endless" I am totally confused, linguistically. I simply do not understand what you are talking about. Infinity is a noun. Infinite is the corresponding adjective. Therefore, the phrase "infinity as endless" means, to me, "the property of having no end" (is) "without end". I do not understand what this phrase is intended to mean. So you are completely correct in your conclusion that "Considering infinity as endless leads to ... confusion(s)". My confusion is linguistic, not conceptual. Not understanding the language, I never get to consider a concept, mathematically or otherwise.

To Wes,

1. Perhaps, but that is basically considering x to be x. It doesn't really say anything.

2. Yes, and many other things, also, positive and negative.

Dear researchers friends,

1, “Infinite” is there in our science not matter from what point of view--------mathematics, psychology, philosophy, linguistics …

2, “Two kinds of Infinite with their different naturals and academic definitions: potential infinite and actual infinite” is also there in our science not matter from what point of view--------mathematics, psychology, philosophy, linguistics, …

Now, 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 “infinite” in our science (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 Wes，

I think if we have enough reasons to have two definitions on “infinite” (such as in terms of “repetition” and “states” for “potential infinite” and “actual infinite”) then we are sure to have enough reasons to have 3 or 4 or more definitions on “infinite” (such as in terms of mathematics or in terms of physics or in terms of kids or in terms of adults or… ) and then, paradoxes follow up as it is impossible to distinguish between or among them in our real operations------it is impossible for people to combine / conflate state and process together and ancient infinite related Zeno’s Paradox and its family member of modern Harmonic Series Paradox are typical examples.

It is a disaster and seems awfully pity that we are still living in Zeno’s time in front of Zeno’s Paradox!

Best wishes, Geng

Dear Wes，

1, the variety of infinities mentioned by me in above post is only purposely for the question from the concept definition theory lever: “can we have two or three or more different definitions with different natures on infinite in our science” but not really for us to choose one from them.

2, I think from the cognition point of view, we have at least two kinds of notions in our science: one is ontological and another is formal (the carrier of the ontology) ------there are no adjectives for ontological ones but large verities of adjectives for formal ones.

Is “infinite” ontological one or formal one?

My regards, Geng

Dear all,

The discussion appears more and more philosophical. Nothing wrong with this in principle. However, properties like redness, questions like "can you see colour" and related paradoxes take you into much more complicated problems, see e.g. the thread on "Is Chalmers' so-called "hard problem" in consciousness real?"

From a practical point many cases involving zeroes and infinities can in most situations be handled by accurate mathematical methods and rigorous limiting procedures.

Dear Wes，

I tried many times to get into https://www.youtube.com/channel/UCyzQtjyWb9itR4dGykpWwKw, but failed.

Could you tell me your opinion to following two questions:

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

2, How can we combine / conflate state and process together to solve the problem of “brackets-placing rule" producing infinite numbers each bigger than 1/2 from Harmonic Series?

Regards, Geng

Dear Wes，

I tried many times to get into https://www.youtube.com/channel/UCyzQtjyWb9itR4dGykpWwKw, but failed.

Wes, what do you think of following two questions:

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

2, How can we combine / conflate state and process together to solve the problem of “brackets-placing rule" producing infinite numbers each bigger than 1/2 from Harmonic Series?

Regards, Geng

Dear Erkki,

Let’s come back to mathematics. How do you think of the “potential infinite--actual infinite” related Harmonic Series Paradox in above post?

Regards, Geng

Geng,

Thanks for insisting! Of course since I understand the math that goes in to Harmonic series paradox or equivalently Gabriel's Horn, I am not too upset even if it appears to be against common sense. Practically speaking comparing volumes, areas and lines will lead to paradoxes if one is not careful.

Take for instance the three dimensional Cartesian space as represented by ellipsoidal coordinates, where the three variables are mapped distinctly on the positive real axis (0,∞). Paradox?

Dear Erkki, thank you for your idea and new example.

> I am not too upset even if it appears to be against common sense. Practically speaking comparing volumes, areas and lines will lead to paradoxes if one is not careful.

Dear Stefan, I agree with your “logical operation of negation” idea in many of our research work, it really helps us to open our minds and eyes in our cognitive activities and brings new materials for our science construction (not only in our science but our daily life as well). I remember when I was young; I asked myself a very naïve question: what is the negation of 0 (ZERO)? So, a new number form (YAN) opposite to 0 was added into number system when I turned my attention to “number spectrum”.

The problem we have now is: infinite related paradoxes were produced because of the confusing of “potential infinite” and “actual infinite”; these paradoxes have been urging us to do something since at least Zeno’s time 2500 years ago. Especially, those modern versions of Zeno’s Paradox (such as Harmonic Series Paradox) tell us a truth: something wrong ontologically or formally in present traditional science system about “potential infinite” and “actual infinite” should be found out and some reformations (big or small) should be done.

Regards, Geng

Dear Wes, thank you for your direct and frank opinions.

1, why “One can but should not describe infinite as small or big because they are not directly comparable”?

2, the problem of “brackets-placing rule" producing infinite numbers each bigger than 1/2 from Harmonic Series has been described with words in addition to mathematical symbols in the above post at page 1 of this thread.

Wes, thank you very much for your new address, I really can get in this morning.

Regards, Geng

Dear Wes, the only feeling for me is thank you for your direct and frank opinions.

1, I believe you use “potential infinite” to explain “why infinite are not directly comparable?” and “actual infinite” to explain “why infinite are indirectly comparable?”, most people do in the same way. But the problem is: our infinite related paradoxes history tells us that in our practical cognitive activities, present traditional “potential infinite--actual infinite” theory system have been being unable us to distinguish whether the things we are treating “potential infinite” or “actual infinite”-------what is indirectly comparable in terms of values or what is not directly comparable in terms of repetition!

2, Harmonic Series Paradox is one of the newly discovered family members of Zeno’s Paradox, it is impossible for you to find it in any present literatures about paradoxes, this modern version of ancient Zeno’s Paradox worries me more because this fact reminds me that we human scientists failed to do anything to solve the defects disclosed by Zeno’s Paradox in 2500 years period of time although people have been trying very hard ever since.

The truth worries me in Harmonic Series Paradox is that exactly same defects disclosed by Zeno’s Paradox.

3, it is unreasonable and unnecessary to be unsatisfactory to Zermelo-Fraenkel set theory, I am unsatisfactory to the working ideas in solving those defects disclosed by infinite related paradoxes at lease from after Zeno’s Paradox.

4, it is ok that 2500 years pass in a flash, I just try what I can in my life. You are a very good researcher friend to me and your post is surely not discouraging me at all, I really appreciate your direct and frank opinions.

Thank you again, Wes!

Sincerely yours, Geng

Dear Miguel, very happy to hear you again, thank you.

Now, do you think we can really 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 Miguel,

So Miguel, how many number items each bigger than 1/2 can people get by “brackets-placing rule" from Harmonic Series?

Yours, Geng

Dear Miguel,  what you say is really true.

I am sorry, but my purpose is to know: what “potential infinite” and “actual infinite” are theoretically and operationally and, is there anything wrong with our understanding and the definitions to “infinite”?

Yours, Geng

Dear Wes and Miguel,

Yesterday I suddenly think about a strange cognitive problem:

Different appreciations, different thinking ways, different working ways, different feelings, different understandings, different universal ethical principle orientation,…, different mental behaviors (different mental sates) of "normal" or "abnormal" persons -----> different mental productions ("acceptable" or "unacceptable" creativities by "normal persons”).

Those "acceptable (at present or in the future)" creativities corresponding between "normal-normal" and "abnormal-normal" may be called positive ones while the others may be called negative ones.

Our history have proved that “value” is a very relative thing; so, many hidden factors of positive creativities can only be observed and manifested by “their related conditions, or rather, by fates” -------because it is only something about human life.

Regards, Geng

Dear Wes,

 The smallest repetition is 1 and not zero.

Hi Wes, thank you, sorry for misunderstanding.

When talk about 0, numbers, infinitesimals, infinite,…, we have “cognitions” relating to many aspects in our science such as philosophy, mathematics, linguistic,…, because we are working in the very fundamental part in our science. So, to say the truth, what you say really make sense. For “cognitions”, we even have Bionics.

In fact you can see many of my contributions in RG are really not only “very mathematical cognitions”.

Sincerely yours, Geng

Dear all,

I think you are missing out on the most important and interesting things! A zero appearing as a singularity can be of "infinitely" (sorry for the pun) many characteristics - an essential singularity, first order, second order and higher order poles.

In matrix language this transforms to a matrix with zeros every where except for ones above the diagonal. Such a zero matrix has a corresponding Cayley Hamilton equation with zero roots of various multiplicities.

If you want to think about it  Gödel is not far away.

Best

erkki

Dear Erkki,

Is the zero you are talking about belonging to the first category of a kind of reference ------- generator, middle, neutral, beginner, origin, marker,…?

My regards to you, Geng

I am talking about a generalization of the number zero to algebraic structures.

Dear Mr. Erkki J. Brändas, thank you very much.

How do you think of following category of 0?

(1) A kind of reference ------- generator, middle, neutral, beginner, origin, placeholder, marker,…,

(2) Absolutely non-existent ------- without numerical value meaning, the negation of being, objectively nothingness,

(3) Relatively non-existent ------- with numerical value meaning, subjectively nothingness, the approximate nothingness, the result of infinitesimal limit , ….

Is the zero you are talking about belonging to a kind of marker in the first category of A kind of reference ------- generator, middle, neutral, beginner, origin, marker,…? Or belonging to a kind of being in the second category of Absolutely non-existent ------- without numerical value meaning, the negation of being, objectively nothingness?  Or belonging to a new category of 0?

My regards to you,

Geng

I have a problem with translating math to words. The meaning of a mathematical concept reflects the use in a mathematical concept. Therefore "zero" could mean all the things you mention above if the circumstances are such. If a paradox would appear we may have yet another "zero" of the Gödel type.

Dear Mr. Erkki J. Brändas, thank you for your frank opinion and idea.

Can you share with us how many "zeros" of the Gödel type do you think we have so far?

My regards to you, Geng

Dear Geng,

Yes in fact I could do this in the following way:

Consider the following symmetric matrix with the elements given by

Qkl= exp{i𝜋(k+l-2)/n}[𝛿kl-1/n]

This is a complex symmetric representation of an n-dimensional Jordan block, which has the well-known canonical form with zeroes everywhere except for ones "above the diagonal".

Jordan blocks are a good way to represent Gödel type singularities, but this is another story.

If you want to know more about Q, I enclose a pdf below.

With best regards

erkki

I see my earlier post to this thread fell victim to my poor proofreading.  The description of Kuratowski finite was incomplete:  leaving out the parenthetical about free join semilattices existing by finitary constructions, it should have read "a set (object in a topos) is Kuratowski finite if the free join semilattice has a maximal element."

Dear Mr. Erkki J. Brändas and Mr. David Yetter, thank you very much.

Sincerely yours, Geng

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-actual infinite”, “infinitesimal with numerical value meaning-infinitesimal without numerical value meaning”,… aroused by the suspended infinite related paradoxes are still on and on endless. Something should be done in this field to improve our mathematics and science.

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 view points.

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 wanting to redefine mathematical ideas. 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，thank you for your frank view points.

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

Best regards,

Geng

Dear Mr. Stefan Gruner，

“Think, for example, about the difference in infinity between countably-infinite (Natural numbers) andover-countably-infinite (Real numbers).”------- “N is less endless, less unlimited and less infinite than R”？

Best regards,

Geng

Geng,

I think that there are some very stark differences between the two.  Both have there place in science and understanding.  There is a big difference between having an assumed infinite and a real infinite. 

In math there is one infinite.  This is the definition of infinite.  A series that has no end. or the list of real numbers or a set of real numbers or a sub-set of anything that goes on numerically forever.  There is always something beyond that can be described by the next number and there is always a next number.  This for all practical purposes is not useful.  It can be shown that counting, or moving through space for an infinite time, or traveling an infinite distance can not be achieved by anyone at any time for any reason.  Therefor it is a hypothetical constraint and has no use in a world with finite beings.

On the other hand understanding the idea of infinite and its implications allows use to develop an understanding of the Universe we must live in and the galaxy which we have a planet that we call home.  It also allows us to look at things differently in that we see somethings as doable and somethings as unattainable.  This means that we can focus on things that can be done with the technology we have today.  The practical side of the sciences which have brought us things like computers, the internet, lasers, TV, cell communications, and on and on.  This is however an understanding of infinite that involves limits.  This is always where people have trouble with understanding.

Limits suggest that we somehow can get past these limits and move on to a faster travel, or a faster computer, or a smarter computer or even smarter people.  Even in our math we put limits on things that in some sense must have a limit so we impose these limits because our understanding is that there "must be one".  In the real world something does not get smaller forever but never get to "Zero".  In the real world something gets so small and then it stops as it has been used up or is no longer there.  We tend to ignore this however when we start developing theories and models of the Universe or Gravity, or Thermodynamics.  

This leads me to our misunderstanding of these issues.  In math there are only limits we impose.  Therefor if we do not look at all things with the discerning eye and an understanding of limits then we will never get to a better understanding of the Universe.  In reality there are only two things that appear to be infinite and that is the Universe and human stupidity.  That is an Albert Einstein quote and I have come close to believing it several times in my scientific studies. 

The idea that a force called "gravity" has no limits is in my estimation the most unintelligent idea that has been thrown on the scientific community.  If this were the case if gravity were a force as it is described the Universe as we know it could not and would not be here.  All the theories that use forces that seem to go to infinite distances or tend to zero but never ever get there are wrong.  This is not how things work it is only how we justify the math.

Our problem with the divide by zero and our thought that it is undefined are the thing that hold us back from developing theories that actually come up with reality instead of things like and expanding Universe or forces that have not end (that one sounds more like a religious dictum than anything else).  Divide by zero is zero, should be what we say.  If it were a summation instead of a division problem we would have the correct answer. 

Everyone understands that we live in this world for a very short time.  We are born, grow up, and die.  These are not things of fiction yet every day we go out into the world and profess things that have no limits in science.   I am not making any religious statements here only scientific ones.  

If I were not here typing theses answers to the question of infinite and its definitions there would be no reason to consider anything but where my next meal is going to come from or if I will survive another season of hunting for food.  However this is not the case some of the creatures on this planet have gotten to the point in Maslow's hierarchy where the basic needs are not the only thing we can think about. 

However there are almost as many on this planet that are on the other end of this spectrum and can not contemplate anything but basic needs.  This is why it is so important to understand why a belief system that allows us to make mistakes that go on for sometime centuries will only slow down our progress and endanger the lives of billions of people that will not get feed because we can not even understand when we must be wrong.

No one will want to feed the world when we believe that there is no end and there is an infinite number of changes to get it right.  There are not an infinite number of changes and our time on this planet is being wasted by people that believe in things that are not real. 

In the Finite wold that we live there are too many resources being used to defend un-defendable theories that survive yet another logical attack from the realists of the world.  In this place people die and lives are lost for no good reason other than we are too busy fighting over theories that must be wrong.                 

Dear Mr. Stefan Gruner and Mr. George E. Van Hoesen，

Thank you very much for your frank point of views.

Our science history proves that it is quite ok that we can have different understandings and different terms for for infinite: “potential infinite--actual infinite”, “assumed infinite -- a real infinite”, “more infinite -- less infinite”….whatever. But the critical problem is in present classical infinite related theory system, those “different understandings and different terms” are unable to tell us “what is what------ potential infinite or actual infinite or assumed infinite or a real infinite or more infinite or less infinite” and how to behave (treat, operate). The newly discovered Harmonic Series Paradox of Zeno’s Paradoxes Families is a typical example: are the items in Harmonic Series belong to potential infinite or actual infinite or assumed infinite or a real infinite or more infinite or less infinite, and how to treat them------this 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.

In fact, the self-confusing theory of “potential infinite--actual infinite” in present classical infinite related theory system describes nothing on “infinite”.

Best regards,

Geng

Geng,

You are correct.  There is no way to tell what is ment by any of these terms.  My understanding is that we must look that the practical answer to this and make an assumtption that there are for the most part limits on the purpose of infinity.

These limits have to be within the realm of what it is we are trying to answer with our understanding.  A line that travels across the galaxy from end to end is on a galactic scale infinite to us sitting on a planet somewhere close to the central part of the galaxy.  To someone looking at our galaxy that line has definite limits.  To an ameba the universe is only the liquid that surrounds the cell that is its being there fore the limit is very small to that organism.     This very practical approach is one that I had hoped was what we were using in science today but it is vary apparent that this is not the case.

If we were to all go back to the first computer class that we took, (hoping that it was a pure logic class and a study in TTL logic) then we would understand the importance of logic and reason in science.  If I have a theory that breaks all the rules in science even for a second then there is something wrong with the theory or the laws of Physics.

We are all fooled into believing the theories of the past are correct even when we know that ignored for them to be correct we had to break some of the rules some of the time.  There is no logic in that.

Dear Mr. George E. Van Hoesen,

Your examples are very reality. In fact, “infinite” can be understood as “something exists in nature world beyond human’s finite ability and something exists in human science within human’s finite ability”; it is ok for most scientists working in applied fields ignore “infinite” in their heart because they need “something exists and can be expressed in human science within human’s finite ability”.

So, “infinite” exists both in nature world and in our science (as a product of human mind). But the problem is: how it exists in nature world and how it exists in our science?

It is very much difficult to break some of the rules some of the time, but this really happens in time out of us human’s will just because of us researchers’ nature and obligation.

Best regards,

Geng

Dear Geng,

When you are talking about infinity in natural world and infinity in science you are talking about two different things. I think that we need some clarification here.

You are talking about the real world and about mathemamatical structures. We have to pose a question: why could we talk about infinity in this two different contextes? Of course, we have some intuition, but in fact very often we assume that in nature there are some structures, which have mathematical properites. In the other words - we assume, that the world is mathematical. This is very strong, ontological assumption and in this case the answer is trivial.

In fact in the science we assume only, that the natural world could be described by mathematical structures. It does not tell us if the nature is "mathematical", it tells us only that we could sucessfully describe the nature by mathematical structures. So in science we are trying various mathematical structures for describing the physical reality, but we could not be sure if they are "proper". Never. Sometimes we could only state that theses structures are fitting the wolrd very good in many cases, but it is all. I think that the questions like "how the infinity exists in nature world?" suppose that there is another way to say something about the properties of physical reality instead of science. The problem is that all a priori approaches were not effective in explaining the natural reality - the modern science is the only way we know to approach the "physical reality". Whatever it is. So if we do not accept that the nature is mathematical we could say only something about the infinity which could be described and modelled by science. This is the infinity from the poinf of view of concrete theory. We could not say anything about the infinity in the nature itself, instead of science.

Best regards,

Paweł

Dear Paweł，

Thank you very much for your frank points of view.

What I say infinity in natural world, I mean the source of infinity in science. Our surrounding natural world is there without any “properties”------- just being and never cares what kind of “knower” visit or stay with it. It is us human who “decide” this or that “properties” for it (such as “mathematical properties”) and different “knower” will “decide” different “properties” for it (such as ants as “knower” or birds as “knower”).

So the natural world can be (1) the necked natural world nothing to do with any “knower” and (2) the natural world being cognized by certain “knower” (such as human).

I agree with you that infinity in natural world and infinity in science are two different things. But the problem are: (1) very often, people confuse “things in necked natural world”, “things in knower related natural world” and “things of knower related natural world in knower’s science”;(2) very often, people confuse “things in knower related natural world” and “things of knower related natural world in knower’s science”-------the “infinite” we are now talking about is a typical case.

Best regards,

Geng

Dear Geng,

Thank you for the answer. Now I think that the problem is more clear.

For me the last distinction is the most interesting. When you want to say something about the world being cognized by a knower and when you accept the hypothetical nature of the knowledge about the "external world" this last distinction is not so clear. From this point of view you could only say about "things described by hypotheses" or about "things described by mathematical/special/scientific hypotheses". "Things in knower related natural world" are the things "showed" by hypotheses, because we are not able to talk properly about anything else.  Of course we could produce a lot of mathematical models without application to the physical reality, and we could analyze their properties but it do not say anything about "our external world" until the application of these models.

So, the fascinating problems with the concept of infinity are connected with the hypothetical (mathematical) structures. Indeed, the question how the people coined  this concept (structure) is very, very interesting - e.g. from the point of view of history of science there are some interesting theological and philosophical impacts on this concept. From the point of view of cognitive science/philosophy of mind it could be formulated as the question about the sources of mathematical intuition (or non-algorithmic nature of our minds). There are many other perspectives... I am curious which aspect is the most important for you. I hope your response will focus the future discussion.

All the best,

Paweł

Dear Paweł，thank you for the interesting thoughts

1, I guess your statement of "things described by hypotheses" or "things described by mathematical/special/scientific hypotheses" may mean more exactly than what I said "Things in knower related natural world", am I right? But I think the "knower" (including some other knower, not only human) can produce something more than “mathematical/special/scientific hypotheses".

2, infinite related paradoxes disclosed tough fundamental defects in present science theory system (including mathematics and philosophy). Many aspects are important in this working field-------this is the very reason why “the black cloud over infinite related science suspended for more than 2500 years”.

Best regards,

Geng

Geng,

I also think in this talk it is important to realize that as humans we have the ability to imagine beyond reality.  This seems on the outset to be trivial but in closer contemplation this is an essential aspect of the talk.  If I have an idea that in know way can be real it may not be seen as unreal by a being that can imagine it being real.  

This ability to imagine beyond reality gives us great understanding in reality but can also lead us in science to theories and points of view that lie outside reality or as Albert Einstein would have said, may "lead us down the garden path", or to unrealistic conclusions.    

Dear George，

Thank you for sharing with us the insightful thoughts. It is true that the fundamental defects in present infinite related classical science theory system are produced by many factors. But science is our human’s, we can find a “garden path" through our hard working and I am sure we are on the way.

Best regards,

Geng

“Potential infinite” and “actual infinite” have been two important concepts in our present classical mathematical theory system at lest since Zeno’s time more than 2500 years ago.

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 endless, they have inborn and numerical natures” and the typical cases are all the ideas and theories of uncountable infinite sets: 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,…

Can we really have many different definitions with different natures for the concept of “infinite” in human science?

Can we name any infinite things “potential infinite” or “actual infinite” or more infinite or supper infinite or… as we like?

Can we pay attention on “potential infinite” and neglect “actual infinite” or vice-versa as we like?

…

That is why we need A Revolution in the Infinite Related Foundation of Mathematics to dispel “the huge black cloud of infinite related paradoxes over mathematics sky”.

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 mapping 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 all the numbers (bijective) 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?

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

Potential infinite was introduced by greek philosopher Aristotle.  Actual infinite was introduced by Cantor in his set theory. I remark that set theory is the mathematics of infinity, other than all other caracterizations given to it. 

Dear Dr. César Rodrigues, thank you.

In fact, our infinite related science history tells us that the idea of “actual infinite” was introduced at least since Zeno’s time when he created Achilles--Turtle Race Paradox 2500 years ago, but Cantor applied “actual infinite” in his set theory.

Do we have “actual infinite sets” and “potential infinite sets” in our mathematics?

Thank you again for your ideas.

Regards,

Geng

“Potential infinite” or “actual infinite” decides our infinite related behaviors in mathematics. The newly discovered Harmonic Series Paradox is a most typical example.

Such divergent proof of Harmonic Series 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）

Each operation in this proof is really unassailable within present classical “actual infinite--potential infinite” related science theory system. But, the unavoidable practical trouble is how many items in （1） can be added up by “brackets-placing rule" to produce the infinite items in （3） each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or…?

“Potential infinite” or “actual infinite” decides whether or not we can produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from Un--->0 infinite Harmonic Series by “brackets-placing rule".

（a）, the confusing and mixing of “potential infinite” and “actual infinite” in present classical infinite related philosophy and mathematics decides we can produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from above operation, but the fact is not------whether Achilles can catch up with the Turtle or not in Zeno's Achilles--Turtle Race Paradox.

（b）, most people in the world now are deeply convinced that we can produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from Un--->0 infinite Harmonic Series by “brackets-placing rule"------not matter how fast Achilles runs, he will never catch up with the Turtle in Zeno's Achilles--Turtle Race Paradox.

（c）, the unavoidable fatal confusing and mixing up errors of “potential infinite” and “actual infinite” in the foundation of present classical infinite related philosophy and mathematics create all the suspended “potential infinite--actual infinite” related paradoxes families------ the 2500 years old huge black cloud of infinite related paradoxes over mathematics sky.

Are “dx--->0 infinitesimal” in differential and “Un--->0 infinitesimal” in Harmonic Series the same things in our mathematics?  

If “Yes”, why we have totally different operations on them? If “No”, what are the differences and how to treat them differently and why?

Lacking systematic cognition to “infinitesimal” by the confusion of “potential infinite” and “actual infinite” in the foundation of present classical infinite related philosophy and mathematics, no one in the world now can answer above question scientifically and this is the very reason for many “suspended infinite related paradox families” in present classical infinite related mathematics.

In present classical infinite related science system, it has been admited that the concept of infinite is composed by both “potential infinite” and “actual infinite”. On the one hand, no one is able to deny the qualitative differences and the important roles “potential infinite--actual infinite” play in the foundation of present classical infinite theory system; on the other hand, no one is able to deny that the present classical set theory is basing on “potential infinite--actual infinite” concepts as well as its related whole present classical infinite theory system. The fact is: any areas in present classical infinite related science system (of couse including present classical mathematical analysis and set theory) can not run away from the constraining of “potential infinite--actual infinite” concepts-------all the contents in present classical mathematical analysis and set theory can only be existing in the forms of “potential infinite mathematical things” and “actual infinite mathematical things”. But, the studies of our infinite related science history have proved that no clear definitions for these two concepts of “potential infinite--actual infinite” and their relating “potential infinite mathematical things--actual infinite mathematical things” have ever been given since antiquity, thus naturally lead to following two unavoidable fatal defects in present classical set theory:

（1）It is impossible to understand theoretically what the important basic concepts of “potential infinite” and “actual infinite” and their relating “potential infinite number forms, potential infinite sets” and “actual infinite number forms, actual infinite sets” are and what kinds of relationship among them are. So, in many “qualitative cognizing activities on infinite relating mathematical things (such as all kinds of infinite sets, elements in infinite sets, numbers of elements in infinite sets)” in present classical set theory, many people even don’t know or actually deny the being of “potential infinite” and “actual infinite” concepts as well as their relating “potential infinite number form, potential infinite sets” and “actual infinite number forms, actual infinite sets”--------it is impossible at all to understand clearly and scientifically the exact relationship among the important basic concepts of “infinite, infinities, infinite many, infinitesimals, infinite sets, elements in infinite sets, numbers of elements in infinite sets”, ... So, it is impossible at all to understand clearly and scientifically all kinds of different infinite sets (such as lacking of the “’set spectrum’ for the overall qualitative cognictions on the existing forms of infinie sets”), elements in an infinite set (such as ”are the infinie related elements potential infinite mathematical things or actual infinite mathematical things, how they exist?”), numbers of elements in an infinite set (such as ”are they actual infinite many or potential infinite many?”), the “one-to-one coresponding theory and operation” in infinie sets (such as ”are the potential infinite elements coresponding to potential infinite elements or actual infinite elements coresponding to actual infinite elements or actual infinite elements coresponding to potential infinite elements?”) ,... --------the unavoidable defects of qualitative cognition on infinite sets and their elements.

（2）First, it is impossible to understand whether the “elements in an infinite set, numbers of elements in an infinite set and all kinds of infinite sets” being cognized in present classical set theory are “potential infinite mathematical things” or “actual infinite mathematical things”, whether there are different theories and operations for “potential infinite mathematical things or actual infinite mathematical things”, and it is impossible at all to understand correctly (scientifically) in present classical set theory the natures of infinite related quantitative cognizing theories and tools (such as limit theory and the “one-to-one coresponding theory”) and their operational scientificities-------- it is impossible at all to master correctly (scientifically) the operational competences and skills of limit theory and the “one-to-one coresponding theory” thus resulting in no scientific gurantee for the operations of limit theory and the “one-to-one coresponding theory”; second, it is impossible at all to judge the scientificities of many infinite related quantitative cognizing activities in present classical set theory, people in many cases can only parrot every bit of what have been done by others or do as one wishes to treat many “not—knowing—what” infinite mathematical things with the unified way of “flow line” (any “infinite sets”, “elements of an infinite set”, “elements’ number of an infinite set” can either be “potential infinite” or “actual infinite”, neither be “potential infinite” nor “actual infinite”, first “potential infinite” then “actual infinite”, first “actual infinite” then “potential infinite”, ,,,), those believed and accepted Russell’s Paradox, Hilbert Hotel Paradox, Cantor’s operations of “cutting an infinite thing into pieces to make different super infinite numbers” and “proving the uncountability of real number set by diagonal method” as well as the famous “applying Russell’s Paradox to prove the Power Set Theorem” are tipical examples of “potential infinite--actual infinite” confusing operations--------the unavoidable defects of quantitative cognition on infinite sets and their elements.

Our studies proved, any the quantitative cognitions to “scientific things” in all science areas can't do without kinds of “scientific carriers” as well as their related quantitative cognizing theories and operations. The fundamental defects caused by the absence of the whole “infinite carriers’ theory” and its related quantitative cognizing theories in present classical infinite theory system exert ill influences on the scientificity of our quantitative cognizing acivities to “infinite set, the existing forms of elements in infinite set, numbers of elements in infinite set”,... in present classical infinite set theory, which inevitablly results in not only the producing and suspending of all kinks of infinite related paradox families in present classical infinite set theory since antiquity but also creating new members of paradox families from time to time, challenging human’s wit and urging us repeatedly to construct scientific (paradoxes free) foundation system for infinite theory and infinite set theory.

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