Dear Dr. Peter Breuer and Dr. Stefan Gruner, thank you for your frank and insightful ideas.

@ Dr. Stefan Gruner, You are right; it is something which I have newly 'uncovered'.

@ Dr. Peter Breuer, the problem is if there are really many different concepts of infinite with different meanings and definitions (such as “Outside Infinite” and “Inside Infinite”), there should be a back up theory system to teach people how to identify different kind of “infinite” or “infinites” for those “infinite things” in our cognizing activities.

According to my studies, at least from Zeno’s time, most people (except Zeno with his creative “actual infinite--potential infinite” related paradox families) have been ignoring a fact that our whole present infinite related mathematics theory system has been based on “actual infinite--potential infinite”. The indefinable and confused “actual infinite” and “potential infinite” have been troubling us human for at least 2500 years with all kinds of debates, magics, paradoxes,… ------ the “2500-year-old huge black cloud of infinite related paradoxes over mathematics sky”.

So, most people nowadays still don’t realize the fact that in our present infinite related mathematics theory system, all the “infinite things” and our infinite related behaviors are unconsciously under the dominating of “actual infinite” and “potential infinite”-------many “infinite related debates, magics, paradoxes,…” have been produced, prosperous.

The exactly same question goes to infinitesimals: are there “actual infinite small (few)” and “potential infinite small (few)” in our mathematics?--------- Take Un--->0 in Harmonic Series and dx --->0 in calculus for example: (1) are they same infinitesimals under the same infinite concept with the same infinite definition? (2) are they “actual infinite infinitesimals” or “potential infinite infinitesimals”?

Best,

Geng

Dear Dr. Peter Breuer, thank you.

Best regards,

 Geng

Dear Dr. Peter Breuer,

“Potential infinite” and “actual infinite” have been in our present classical infinite related science system since antiquity------dominating how all the “infinite things” exist in our science and how we human behave in our cognitions to all the “infinite things”. But, some say “all the infinite things in mathematics are potential infinite and they are treated by set theory and limit theory” while some others say “all the infinite things in mathematics are actual infinite and they are treated by set theory and limit theory”.

The confusing and debating on “potential infinite and actual infinite” are hard and endless at least since Zeno’s time 2500 years ago. So, we can see many suspended infinite related magics, paradoxes; and, we have many different ways freely creating “more (much) infinite, more more (much much) infinite, more more more (much much much) infinite,… in our mathematics”: power set; cardinality aleph one; cutting an “infinite thing” into pieces by “the first generating mechanism, second generating mechanism, third generating mechanism,…” to produce super infinite, super super infinite, super super super infinite, super super super super infinite, …; ...

In fact, the situation is: to many mathematicians, “the infinite related number forms are invisible and untouchable variables, they are not numbers” although they really see them, touch them and treat them as numbers very often in calculus while to many people who are working in practical science (such as physicists), they are brave enough to say “the infinite related number forms are visible and touchable numbers”.

The most trouble thing for me in such confusing and debating on “potential infinite and actual infinite” is: when facing and treating the infinite related number forms, we have to declare first that they are not numbers (such as infinitesimals and infinities) theoretically in mouth but tell in mind firmly for those very infinitesimals and infinities “forget what have been just said in mouth and they are in fact numbers other wise no practical numerical operations can happen”.

When I first come across such operations 40 years ago I feel very sorry for myself that I have been very often as so many people played the role of a minister in Andersen’s fairy tale The Emperor's New Clothes...

Best,

Geng

Dear Dr. Stefan Gruner, thank you very much for your historical view points on “actual infinite”, “potential infinite” and "our mathematics ".

I agree with you that the debates on “actual infinite and potential infinite” in both philosophy and mathematics have a long history and are still endless so far.

Now the problem is clear that if we acknowledge both “actual infinite” and “potential infinite” in human science, these two different concepts are there and dominate things in both philosophy and mathematics-------unavoidable. So, we have to ask things in philosophy or mathematics we meet: are they “actual infinite or potential infinite”? If these two different concepts are confused, our behaviors on infinite things will surely be confused-------paradoxes are created (such as “can infinite items each bigger than 1/2, or 100, or 1000000, or 1000000000000000,…really be produced by the ‘brackets-placing rule’ from Un--->0 Harmonic Series”?).

Thank you again dear Dr. Stefan Gruner.

Best regards,

 Geng

Dear Dr. Peter Breuer , thank you.

I agree with you that there can be many leisure talks after tea time and dinner time on infinite.

Now the problem being focused on is not “leisure talks”: if we acknowledge both “actual infinite” and “potential infinite” in human science, these two different concepts are right there and dominate things in both philosophy and mathematics-------unavoidable. So, we have to ask things in philosophy or mathematics we meet: are they “actual infinite or potential infinite”? If these two different concepts of “actual infinite” and “potential infinite” are confused, our behaviors on infinite things will surely be confused-------paradoxes are created.

Thank you again dear Dr. Stefan Gruner, but what do you think of following practical mathematical question: how many items each bigger than 1/2, or 100, or 1000000, or 1000000000000000,…can really be produced by the ‘brackets-placing rule’ from Un--->0 Harmonic Series”?.

Best regards,

 Geng

Dear Dr. Peter Breuer ,

1, I really agree with you that there are important concepts of Euclidean and non-Euclidean geometries, but we are not confused on them; so, no confused behaviors and paradoxes are created.

2, It is nothing to do with convinces but a very practical mathematical question: how many items each bigger than 1/2, or 100, or 1000000, or 1000000000000000,…can really be produced by the ‘brackets-placing rule’ from Un--->0 Harmonic Series in present classical “actual infinite” and “potential infinite” related mathematics (human mathematics but not my mathematics)?

My heartfelt thanks to you, dear Dr. Peter Breuer, your view points are really meaningful and appreciated.

Best regards,

 Geng

There may be as many different theories of infinitesimals as we like, but the problem is: because of the confusing “actual infinite and potential infinite”, none can offer a non-paradoxical answer to the above very practical mathematical question raised by Harmonic Series in present classical “actual infinite” and “potential infinite” related mathematics.

So, something must be wrong in present classical “actual infinite” and “potential infinite” related philosophy and mathematics!

Dear Dr. Peter Breuer , thank you.

@ whatever you mean, the answer is surely "as many as you like (but not too many) so not even you can see a `paradox' there " in your above post.

Do you think that is the mathematical ways you are so sure for the reason “there is no confusion. And nobody has the faintest idea what you mean by the `Harmonic series', nor what you mean by `Un' nor what you mean by 'brackets-placing rule'”?

I agree with you that we really have so many infinite related theories in human science and mathematics since Zeno’s time, but according to my studies, none of them can offer a non-paradoxical answer to the above very practical mathematical question raised by Harmonic Series. And, the very Harmonic Series is really right there in our human mathematics.

So, something must be wrong in present classical “actual infinite” and “potential infinite” related philosophy and mathematics!

Best regards,

 Geng

Dear Dr. Peter Breuer , thank you for your kindness.

1, Basing on the 40 years studies of the suspended infinite related paradoxes and the fundamental defects and errors in present infinite related science theories; I am very sure arguing “something must be wrong in present classical “actual infinite” and “potential infinite” related philosophy and mathematics”.

Would you please tell me why you are so sure to say “Deduce that your studies are wrong! Or your logic is” in your above post?

2, I would like to know more exactly and clearly what question I didn’t answer you about Harmonic series?

Best regards,

 Geng

Dear Dr. Miguel Ángel Montes , thank you for your frankness and let me be frank, too.

Let’s come back to see following divergent proof of Harmonic Series which can still be found in many current higher mathematical books written in all kinds of languages, which is a typical example of confusing “potential infinite” and “actual infinite”:

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 the “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".

1, 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.

2, 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 10000000000000 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.

3, the 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 “potential infinite--actual infinite” related paradoxes families------ the 2500 years old huge black cloud of infinite related paradoxes over mathematics sky.

4, being frank, I am seriously ask you a very practical mathematical question: how many items each bigger than 1/2, or 100, or 1000000, or 1000000000000000,…can really be produced by the ‘brackets-placing rule’ from Un--->0 Harmonic Series in present classical “actual infinite” and “potential infinite” related mathematics (human mathematics but not my mathematics)?

5, no one in the world within present classical “actual infinite--potential infinite” related philosophy and mathematics can answer such question scientifically at least since Zeno's time 2500 years ago. So, I am very sure arguing “something must be wrong in the foundation of present classical “actual infinite--potential infinite” related philosophy and mathematics, a new infinite theory should be pioneered and developed”.

My heartfelt thanks to you, dear Dr. Miguel Ángel Montes, your frank view points are really meaningful and appreciated.

Best regards, 

Geng

Dear Dr. Stefan Gruner  , thank you for your view points.

1, I agree with you that many people think it is obviously “of course the running Achilles will overtake the walking turtle within a finite period of time”, even a middle school student can tell from the simple calculation how much time the running Achilles needs to overtake the walking turtle in a real situation; still some one demonstrates the on site race with an alive walking turtle to prove Zeno's Paradox is pseudo. But still many more others disagree with such opinions and proofs. Ok, that us regard it as a “leisure talks”, but how about a very practical mathematical question (not a “leisure talks” at all): how many items each bigger than 1/2, or 100, or 1000000, or 1000000000000000,…can really be produced by the ‘brackets-placing rule’ from Un--->0 Harmonic Series in present classical infinite related mathematics.

2, I agree with you that there are many “finite-infinite” stories in our science such as “Mandelbrot Set”, “a piece of finite line and the infinite points within”,…. Are they “actual infinite sets” or “potential infinite sets”?

Best regards, 

Geng

Potential infinite was a concept defined by Aristotle. Actual infinite sets were introduced by Cantor in set theory, the mathemetical science of infinite.

Dear Dr. César Rodrigues, thank you for your view points.

“Potential infinite” and “actual infinite” have been tied together with “infinite” as the foundation in our present classical infinite related science system since antiquity------dominating how all the “infinite things” exist and how we human behave in our cognitions to all the “infinite things” in our science.

We have the concept of actual infinite, so, “actual infinite small (few)” and “actual infinite big (many)” were unavoidably introduced into mathematics. We have the concept of potential infinite, so, “potential infinite small (few)” and “potential infinite big (many)” are unavoidable in mathematics.

Now, the problem is: the confusing of “actual infinite--potential infinite” has produced suspended and unsolvable “infinite big (many)” and “infinite small (few)” related paradoxes in set theory and mathematical analysis theory in present classical infinite related mathematics.

Best regards, 

Geng

It is because of the fatal mistakes and defects in the cognizing and definition of “infinite” since antiquity in present classical science system that result in the generating of the two contradictory and confusing concepts of “potential infinite and actual infinite”. So, the following unavoidable and suspended problems and paradoxes have been accompanying us ever since: Zeno’s Achilles--Turtle Race Paradox more than 2500 years ago; Berkeley’s “Zero-Non Zero, Number-Non Number Paradoxes of Infinitesimals” more than 300 years ago; Russell’s Paradox; the newly discovered Harmonic Series Paradox; Cantor’s “four typical errors in the diagonal-contradiction proofs on the uncountability of real number set; …. With different forms of “infinite big, infinite many, infinite small, infinite few, super infinite, transfinite, transfinite numbers, …”, from different angles and in different periods of time, these problems and paradoxes have disclosed the “qualitative-quantitative” cognizing obstacles created by the confusing contents of “potential infinite and actual infinite”.