Convention says "higher" and "lower" infinities, but convention can be changed.

Dear Mr. Sudev Naduvath and Mr. Pedro Montoto García,

Do you think that the uncountable infinite set has more elements than countable infinite set? So, is it logical to say “uncountable infinite set is more (higher) infinite” and “countable infinite set is less (lower) infinite”?

Regards.

 The relationships between infinite sets of different cardinality is studies in the set theory field of Cardinal Numbers (see, as a starting point the wikipedia entry http://en.wikipedia.org/wiki/Cardinal_number)

Dear Mr. Guido Governatori, thank you.

But, can we say “real number set” has more elements than “nature number set” in present set theory?

Regards.

That is true dear Geng,

It is a topic in set theory and cardinality of sets. Infinite sets are not all the same when it comes to measuring sizes.  There are infinite sets that are infinitely more than some infinite sets. The set of natural numbers is a countable set of the smallest infinite cardinal called aleph null. But by Cantor's  theorem of strict  inequality between the cardinality of a set and that of its power set, we see that the cardinality of the set of integers is strictly less than the cardinality of its power set. Then the  continuum hypothesis provides the cardinality of the power set of the set of integers to be exactly that of the cardinality of the set of real numbers or the continuum denoted by script c or aleph one. But the set of real numbers is uncountable and therefore its cardinality aleph one or c  is  the smallest size of an uncountable set. Taking powers of such uncountable sets we get more sets of higher uncountable cardinals than aleph one and aleph two etc.

By the way the infinity (the sleeping eight ) we use in calculus or daily algebra is just a symbol for a very far distant and unreachable place of the reals which has nothing to do with the size or cardinality of infinite sets we are discussing here.  

Dear Mr. Dejenie A. Lakew, thank you.

“More infinite” and “less infinite” characteristics for the present understanding (definition) of “infinite” offers people a great deal of freedom in our infinite relating operations in mathematics since antiquity.

Regards.

You can use standard set theoretic operations on infinite sets, and as usual, and the set of natural number is included in the set of real number (or alternatively one can say that there is a total injective function from the natural numbers to the real numbers (and this is usually taken to indicate that one set is a least as large as the other).

But now the problem we meet is: is it logical to say “uncountable infinite set is more infinite than countable infinite set”----------“Infinite A is more infinite than Infinite B”?

Dear Miguel, happy to see you again.

1, How could we understand “some infinite things have more elements than the other ones------what is infinite”? Can we really run away from hierarchical structure here?

2, why 'higher or lower'? From numeric point of view in mathematics, are they different from 'longer or shorter', 'more or less', 'farther or nearer', 'bigger or smaller',…?

In fact, we come to our cognition of “Infinite”. We have at least two definitions of infinite here: first definition of infinite (Infinite A) is “endless, ever lasting,…”; comparing to the first definition of infinite, the second definition of infinite (Infinite B) is “not as endless and ever lasting,… as the first one”.

 In mathematics, Infinite Set A has more elements than Infinite Set B (more endless, ever lasting,…) so Infinite Set A is more infinite than Infinite Set B.

Best regards,

Geng

Dear Akira，so happy to see you again.

I agree with you that the set of even numbers have exactly the same number of elements as the set of all odd numbers, and the set of all natural numbers is an infinite set according to the definition of infinite in our science: “endless, ever lasting, …”.

Best regards,

Geng

My dear colleagues,

According to new studies, “the infinite law” and “the carriers of infinite law” are confused and mixed up and “the carriers of infinite law” are neglected even unaware since at least Zeon’s time. This “half absence defect” phenomenon does not only happen in set theory but in the field of infinitesimal relating analysis as well------anywhere in our science (especially in mathematics) relating to “infinite”, this is the real source of many infinite relating paradoxes.

 “The infinite law” is the nature, quality, the definition of infinite which are not touchable, visible and quantified while “the carriers of infinite law” are much humanized scientific things which are touchable, visible and can be quantified.

Best regards, Geng

Question topic is directly corresponds to problem of continuum hypothesis (Aleph0

Cantor really contributed a lot in set theory, especially in his work of “infinity quantified”. But the real problem is: he didn’t know why he can do that scientifically. It is not Cantor’s faults but the deep rooted defects in the foundation of infinite related science system since at least Zeon’s time.

Dear Miguel,

1, do you really think 'infinite doesn’t mean infinite’?

I really think that there are 4 mistakes in Cantors' diagonal proof of the no numerability of reals and I had my ideas in the paper of 2010 ICM. I think I am now working on the new foundation of infinite related area of mathematics to get rid of those long being defects disclosed since at least Zeon’s time

2, how we divide the degree of infinite in present set theory in mathematics? The fact is by more or less elements. The infinite elements (quantity) define infinite nature (quality). So, it is true that 'infinite means infinite'. It is not the matter of choosing correct terms for the different degrees of infinite, we can not define different infinite nature (quality) with different quantity just because we can not have thousands of infinite nature defined by different quantity. It is impossible to choose thousands of different terms for different infinite nature defined by different quantity.

3, the subset [0,1] of the real numbers is sure to have infinite elements in amount (ever lasting, endless,…) even though it has a beginning and an end.

Best regards, Geng

You can injectively embed N in R but you cannot injectively embed R in N. So you can say that """R is a bigger set than N""". The other proposed statements "R is more infinite than N" and "R has more elements than R" seems to be both of them wrong, and the first one is worse than the second one. The first one is bad because they are both infinite, something cannot be more infinite than something different. The second one is bad because "more" and "less" makes sense maybe only if one of the two quantites to compare is finite. OK, the second one can maybe work, but this becomes a question of English, and not of mathematics.

So far I would say, in my opinion, the answer(s) by Professor Kanda, University of Toronto, is the 'most interesting.'

Astro Physics and Mathematics:Thesis: The number of elements in the universe is a proper finite subset of the natural numbers by cardinality.

The thesis can be refuted by a counter example.

Dear Mr. Mihai Prunescu and Mr. Ramon Quintana,

Why can we say """R is a bigger set than N"""? Because "Infinite R has more elements than Infinite N"; the truth here is: "more" and "less" makes comparing sense in mathematics (not just making comparing sense in English) for the quantities of two infinite things (Infinite R and Infinite N) but not finite things.

When we enter into the “Infinite” relating field (set theory, analysis,…), we are facing a huge working field: what is infinite? what is infinitesimal? what is infinite-big? what are potential-infinite and actual-infinite? what is mathematics? what is science? what is logic? what is philosophy of mathematics? …

It is now the right time for “philosophy of mathematics” to wake up and work.

Best regards,

Geng

Yes! Look up 'cardinality' of sets at

http://en.wikipedia.org/wiki/Cardinality

http://en.wikipedia.org/wiki/Cardinality

More on 'useful infinite.'

Did any of you watch the recent TV presentation on NOVA about the question 'Is God a Mathematician'? This can pertain to a discussion of what might be coined as 'useful infinite.' With useful infinite (idea) the present complexity of the countability or uncountability is related to whether or not there is a mathematical model (e.g. Maxwell's Equations, Kepiers (extended) Laws, Nobel Laureate - Higgins work- on using a mathematical model) to accurately predict or estimate the actual physical existence of not yet discovered elements or particles in the universe. Fields such as astrophysics, electricity and magnetism, particles physics, etc. offer examples into large but finite. However, the currently understood complexity of weather, macro evolution and others make it presently (not yet) probable to accurately predict or estimate using a mathematical model. The current demise or human intelligence problem is related to work of another Nobel Laureate, the late Herbert Simon much of whose work is based upon his invented idea of 'Bounded Rationality.' This theory, if you will, addresses the limitations of groups of human and their intelligence or capacities in mathematical modeling. I posed Bounded Rationality in an earlier discussion.

I invite your opinions, assertions, conclusions and other examples.

Dave

This issue can be viewed from the perspective of Philosophy of Nature. What is Nature? Is it a given state of affairs or is it a (huge) set of possibilities? Assuming the second alternative, the Nature number set is a potential one. If the real number set is considered (as I do) to refer to (a) potential - not an actual - infinite(s), then the Nature number set and the real number set are...identical in all aspects.

Dear Mr. David C. Rine,

Thank you for your recommends. I hope the idea of 'useful infinite' can be developed into or belong to a theoretical system.

Best regards, Geng

Dear Migue,

We have two cognitions for “apple”: one is the abstract concept which exists in our mind as the abstract picture, the abstract imagination (the abstract apples which is not eatable, not touchable,…)while another is the carrier of the abstract concept which can be different kinds of real objective things (real objective apples which are eatable, touchable,…). “The abstract concept” and “the real objective carriers of the abstract concept” can not be confused and mixed up,

Our science history tells us that in our present traditional infinite system, the abstract infinite concept is called “potential infinite” while the real objective carriers of the abstract infinite concept is called “actual infinite”. Now, the trouble is “the abstract infinite concept” and “the real objective carriers of the abstract infinite concept” have being confused and mixed up, “the infinite concept carriers’ theories” being neglected even unaware since at least Zeon’s time …This “half absence defect” in our present traditional infinite system failed many of our “actual infinite quantifying work (such as infinite sets comparing work and infinitesimal calculations)”

My best regards, Geng

Dear Mr. Akira Kandaand Mr. Alfredo Pereira Junior ,

It is really a very complicate situation in the infinite field of mathematics. When we enter into the “Infinite” relating set theory, analysis,…, we are facing a huge working field: so many people have been contributing in this field. What is infinite? What is infinitesimal? What is infinite-big? What are potential-infinite and actual-infinite? What is mathematics? What is science? What is logic? What is philosophy of mathematics?…

My best regards, Geng

Geng,

The questions you raise are at the heart of the problem in science and math today. 

We all think we know the definitions to all these terms but we all have different ideas about what they mean.  We must have base terms that everyone understand the same way or there is no hope of having the problems of the day answered in a meaningful way.  

I see this in the definition of empty space...  Most will say that there is nothing in it.  But in the back of the scientific mind of many they think that can not exist because they think they know that there are fields in empty space and then there are quantum fluctuations and on and on.  So for science there is no real definition for empty space, no infinity, and no reason to believe that we know what we are talking about.

If we say empty we mean without anything at all...  (no quanta no fields absolutly nothing at all)

If I say infinite I mean it never ends....  (not just until I run out of time to count)

If we would start from these points we could build reality from the start not try to build it from the middle somewhere that is random.

Math is one of the biggest problems with this.  Math is not nature and it has the flaw of being perfect unless we count wrong.  However nature is not perfect and has very noticeable limits.  Our math has a hard time dealing with this and we force it to approximate what we think is correct.  This means we impose reality on the system that we use to measure nature.   Is it any wonder that we have not found the real answer to why there is not a unified field theory! Even the way I posed that question indicates how far we are off base.  Everyone I talk to thinks that the unification of physics implies that it "MUST BE a field equation" that comes out as the answer.  If we look back at the math the thing that should be notable is that the math are all just approximations to reality.  IF Kurt Godel were alive today he would laugh hard at this one.  How can an approximation be considered reality?

Math is the problem not the answer. 

Dear Mr. George E. Van Hoesen, thank you for your insightful idea: “We must have base terms that everyone understand the same way or there is no hope of having the problems of the day answered in a meaningful way.”

I agree with you that science is our human’s, it is undoubtedly that we really can own (create) our common “scientific base” for our science through studies, discussions, …. This is one of the reasons that we are here and we cherish this destiny that ties many of us together.

 How to understand “approximate” and make  it acceptable in our science is really a very tough work.

. My best regards, Geng

Dear Mr. Ramon Quintana, thank you for your kind remind, I really misspelled the word and I have put the right one “natural” in the phrase.

Your ideas are very philosophical and this touches the bottom of “human science”-----how to unify “objective world” and “subjective world” has been one of the toughest works for scientists.

Mr. Ramon Quintana, to say the truth, I believe cognizing and translating “things in human science and things in natural world”, “sets in mathematics and sets in natural world”, “infinitesimals in mathematics and infinitesimals in natural world”,… are essential and important work for scientists though very difficult.

My best regards, Geng

Dear Mr. Ramon Quintana, thank you!

I sincerely hop we can benefit from our discussions here and contribute something for our science.

My best regards, Geng

Our cognition defects to “infinite” and the confused--mixed up of the “potential infinite------ abstract infinite concept, and actual infinite ------real objective carriers of the abstract infinite concept” have been produced ill influences on our studies and cognitions to the foundations of limit theory and infinite related number theory thus unable people’s qualitative and quantitative studies and cognitions to the “actual infinite ------real objective carriers of the abstract infinite concept (sets, infinitesimals, infinities, …)”.

Outside of the Mathematics Theory we have been discussing, let us turn to a more tangible question/problem: Present to me an empirical test that validate that the number of elements in the universe is either countable finite, countable infinite or uncountable. Hubble suggests that the known part of the universe is at least 15 billion light years in expanse and the elements include those which have recently been discovered. With such a test one could expand the idea of 'infinite' to an area other than mathematical curiosity. You can also plea the effect of Nobel Laureate Herbert Simon's Bounded Rationality if you like.

Dear Mr. David C. Rine,  

Will the “known part of the universe” be changeable conceptually and practically?

My best regards, Geng

If there is a 1-1 onto (bijective) function map one set to another set we say the number of elements of two sets are equal. For example in analysis we can show that there is 1-1 onto function from open interval (0,1) to real line, in fact (0,1) is a proper subset of the real line. There are a lot of proper subsets of infinite set which has the number of elements equal to the supper set.

In my opinion, real number and natural number are infinite sets we can not say “real number set is more infinite than natural number set”.

Akira,

I agree with your assessment.  QM is needing to be overturned.  Albert Einstein tried for 30 years but he was to stuck in the math to see the problem.  There is no duality in QM or in the real world.  The problems have been brought on by the thought that you can have your cake and eat it too.  I say that in jest but the implication is that physics has been living in a fantasy world where what ever you can imagine to be real is real at least to the people that think they know everything.

I live in a real world and work in a real lab (even though it is very small).  So when I see something that does not fit in the QM model I have no problem disregarding the old understanding and using what I know to work. 

We must move beyond thinking that a theory that was made even before we knew about the Neutron in the atom can realistically model reality.  YES, the model does help us get to a better understanding, YES it has merit in calculating things we do not totally understand.  Remember that more than one hundred years ago we also disregarded the work of Albert Einstein in favor of Newton only to reconsider and find that his "approximation" of gravity was much better at the scales we needed for planetary motion.

It is once again time to reconsider a better understanding of the universe and move to a new reality in science.  Quantum Mechanics is only the latest step to understanding reality and in my estimation is is one that has stopped us from moving closer to reality.  

I do not as Albert Einstein use to say, "Believe in spooky action at a distance".  Just because we can not measure something very small does not mean we can not understand it or that it is in some way not real.  Things do not just disappear, reappear, hit and then disappear again and then not exist.  We stopped using common sense in this part of our science and stopped looking for the real answer to the problem.  

If we can show that uncertainty is caused by things that are real, then we can correct the course of science and move to better understanding.   

 Dear Mr. Wiwat Wanicharpichat

Why we say mathematically something is more or less, bigger or smaller … than the other （different grade）? One of the main scientific reasons is because of different quantities (not being bijective).

So, just by different quantities, it is logical to say mathematically “real number set is bigger than natural number set”; “real number set is more infinite than natural number set” ….

My best regards, Geng

Dear Geng Ouyang,

How to use the symbol "=" in mathematics?

1)  Let  A  and  B  be sets. If A ⊆ B and B ⊆ A then we have A = B.

2)  Let  f  and  g  be functions from set X to set Y say, such that f(x) = g(x) for all x∈X, we have  f = g; two function are equal.

3) If there is a 1-1 onto (bijective) function map set P to another set Q we say the number of elements of two sets are equal, and denoted by |P| = |Q|.

     Two finite sets of same number of elements, we alway have a bijective mapping from one to another. 

    We can not compare the number of two infinite sets (as Miguel Ángel Montes comment)  "they are not more or less infinite". If we have a bijective mapping from one to another infinite set, the two sets are equal number of elements.

Dear Mr. Wiwat Wanicharpichat

It is generally accepted that the quantities (elements) of two infinite things can be compare so we have the uncountable proof by Cantor for “real number set”------- real number set has more elements than natural number set.

So, different quantities (elements) make different grades of infinite: big-small, more-less, low-high, long-short ….

My best regards, Geng

Dear Geng Ouyang,

Thank you so much for your kindly comments and advice.

Dear colleagues,

Anyhow, physics seems closer to real world than math. There must be very close relationship between scientific ideas in our mind and objective world but this fact is always confused ------- scientific form but objective base.

We see flour and rice on the storage rack in grocery stores but we understand very well that the grocery stores never grow flour and rice.

Akira,

Many people that would put you or I down for our thoughts should consider what side of history they want to be on.  The past one hundred years science has been trying in vain to solve the problems with (Math, Physics, Chemistry, Biology,and more) by using the old thinking.  Einstein once said that "You can not fix the problems of today by the thinking that put us in the situation"  There is no way to fix a broken system by using the broken system. 

If todays scientists want to solve the problems there must be new thinking.  We have had one hundred years of the old thinking and still we are no closer to the solution than when Einstein said "I cannot believe that the great one plays dice".  He is referring to a God that is not all about chance and probability.  I am not going to comment on that except to say that there is an order to the universe that we are missing and it is highly ordered in localized areas and this is the area that life springs up from.  This order does not seem to be random and has a specific path.  Our goal as scientist should be to help us understand the path that science is showing us and to disregard the paths that lead us to dead ends all the time.

I think that one hundred years of dead end is much more than any person should have to put up with and about one hundred years to long for a person or society that claims to be scientific in nature. 

Dear Mr. George E. Van Hoesen and Mr. Akira Kanda,

1, “There is no way to fix a broken system by using the broken system.”

2, “If today’s scientists want to solve the problems there must be new thinking.”

My hard working experiences in the infinite relating area prove what you say above. I totally agree with your frank and insightful ideas.

I am on the way to solve the disclosed defects in the infinite relating area and I will try my best in the rest of my life.

My best regards, Geng

Geng,

Thanks.  It takes all of us working together and in parallel to move beyond the old.  I will for the rest of my life choose not to except mediocracy in any field of science.

Akira,

Very interesting idea. Has this affected you in the past? How would like like to propose some changes? I am looking back to find would for what RG stands. Help. 

Following up a bit more. Are you all (above) suggesting that in, for examples, areas such as the Life Sciences there is a philosophy of science(s) underlying the more dominant culture such that if this philosophy of science(s) is not adhered to there is an underlying bias towards work of those on the 'outside'?

Replace Life Sciences with Physical Sciences. Is the same idea, philosophy of science(s), prevalent?

 Would any of you from the above mentioned dialog wish to share examples that illustrate these biases? That might be interesting and it might help to improve peer review processes.

Best wishes.

Dear Mr. David C. Rine,

I am very sorry that you may misunderstand us. What we focus on in our discussions is in the infinite relating science branches and areas------the defects really have been there since Zeon’s time. I am complaining about the long being mistaken working train of thought because the modern version of Zeon’s Paradox------Harmonic Series Paradox is discovered which proves that we should have new working train of thought to get rid of those disclosed defects being existing since Zeon’s time in the infinite relating science branches and areas.

Sincerer yours, Geng

The discussion, practically, may be reduced to approximations. Just in case do not recall UP, this is Heisenberg's Uncertainty Principle, and UP is about classic limits to physical measurability.

In quantum mechanics, the uncertainty principle, also known as Heisenberg's uncertainty principle, is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle known as complementary variables, such as position x and momentum p, can be known simultaneously. Introduced first in 1927, by the German physicist Werner Heisenberg, it states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.[1] The formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard[2] later that year and by Hermann Weyl[3] in 1928:

(ħ is the reduced Planck constant in the attached formulation).

In applied mathematics one  way of relating infinite to finite is through approximations or estimates as is the case with infinite series and infinite sequences such as those used in solving DE's. How accurate is accurate enough, close enough, and so forth. This relates mathematical theory to engineering mathematics which I use.

More questions. Are you saying that Higgs mathematics is wrong (instead of right) and are you saying that Higgs particle experiment at CDERN which is related to his winning the Nobel Prize is also wrong? If you believe it is wrong can you or someone offer a counter set of mathematical formulations and offer a possible alternative experiment? I understand your concern about experimental repeatabilty. Yes.

What am I missing? 

http://en.wikipedia.org/wiki/Uncertainty_principle

Dear Mr. David C. Rine, thank you for your frank and insightful opinions.

You write “In applied mathematics one way of relating infinite to finite is through approximations or estimates as is the case with infinite series and infinite sequences such as those used in solving DE's. How accurate is accurate enough, close enough, and so forth. This relates mathematical theory to engineering mathematics which I use.”

I really agree with you that in a physician’s eyes, approximations or estimates is enough (applied mathematics or engineering mathematics). But on the other hand, applied mathematics or engineering mathematics is based on theoretical mathematics with mathematical logics.

Sincerer yours, Geng

Hi Akira, 

I am sorry to see you so frustrated and understand your reasons for it. 

I have a question, though. Where, in your understanding, does logic and mathematics in general originate from? In other words, are you a constructivist, intuitionist, formalist, platonist or someone else. I believe that the answer to this questions could pinpoint a source of your frustration with physicists who regard themselves as realists. 

The question I pose above could also be directed to all other participants of this discussion about real or perceived problems with infinity. 

Dear Mr. Akira Kanda, very insightful, thank you.

Well, in present traditional infinite relating science, we have been in fact facing 2 troubles at least since Zeno's time:

1, unable to define what “potential infinite” and “actual infinite” conceptually (theoretically) are, so people have been arguing and debating at least since Zeno's time and will continue endless in present defected infinite relating science system------both parties are impossible to know what they against and what they are for.

2, unable to identify those things being treated “potential infinite” or “actual infinite” operationally (practically), so members of infinite relating paradox families have been born one generation after another. For example, we have Zeno’s Paradox of “Achilles--Turtle Race” 2500 years and now we have a “strict proven” modern version of ancient Zeno’s Paradox------ Harmonious Series Paradox.

It is great we are now here on the way doing something for our science, those disclosed defects should be got rid of sooner or later by us human in our human science.

With regards, Geng

 Dear Miguel, thank you for your kind and frank opinion..

According to my studies, if we really known what “potential infinite” and “actual infinite” conceptually (theoretically) and operationally (practically) are, we would not have the “strict mathematical proven” modern version of ancient Zeno’s Paradox------ Harmonious Series Paradox.

With regards, Geng

Dear Akira,  

Re: “It is astounding that today's "successful" young researchers have 30 things to say per year. I wonder when do they think.”

 Well said. On the other hand, this is what is expected of mathematicians (and researchers from other fields) today. All this is good for business of “educating”: it is highly democratising so everybody who can skilfully quote others can now be a scientist. Unfortunately, critical thinking in this postmodern world is not encouraged nor expected.  

 In your reply to my question, you mentioned what you have been doing instead of “Where, in your understanding, does logic and mathematics in general originate from?” I know that most practicing mathematicians (and other scientists) do not care about such issues anymore.

Dear Miguel, thank you.

But l beg your pardon letting me introduce my frank opinion on this issue.

Ancient Zeno’s Paradox of Achilles--Turtle Race and the “strict mathematical proven” modern version of this paradox (Harmonious Series Paradox) draws us attention on “potential infinite” and “actual infinite” numerically (nothing to do with how a set is built). In such numerical cases, we are unable to identify those things being treated “potential infinite” or “actual infinite” theoretically and practically.

Are the Un--->0 items in Harmonic Series “potential infinite” or “actual infinite”, how to treat them? No one in the world now can tell scientifically whether or not we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite infinitesimal items in 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.

Harmonious Series Paradox is a visible and touchable “Potential Infinite and Actual Infinite Paradox” in front of us. This is the biggest reason for my statements about the lack of theoretical and practical cognitions of 'potential' and 'actual' infinite in present science theory system.

Sincerer yours, Geng

Dear Miguel,

1，set theory or what ever, and forget about the word “items”, let’s come to a very practical and unavoidable question: can we produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite 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.

2,  for the phrase in your post "Are the Un--->0 items in Harmonic Series “potential infinite” or “actual infinite”, how to treat them?", the items are items, not infinites. The problem comes when you try to consider all of them together, and is the way you choose to consider their infinitude, as 'being done' (potential) or 'already done' (actual), what links to each kind of infinity. What are those Un--->0 items in Harmonic Series?

3, “actual and potential infinity are psychological concepts, not mathematical one, so trying to treat them as mathematical will lead you to some mistakes”. If what you say is true, a hundred people can have a hundred understandings of “actual and potential infinity”.

4, my studies on the project of present traditional infinite relating theory system proves me a conclusion: since Zeno's time, we have lots of “infinite theories”, but all of them can do nothing in front of Ancient Zeno’s Paradox of Achilles--Turtle Race and the visible and touchable “strict mathematical proven” modern version of this paradox (Harmonious Series Paradox).

People can keep making whatever “infinite” understandings they like or try to create different formal languages for such understandings under present traditional infinite relating theory system. But if these understandings and formal languages can do nothing to solve the practical problems as well as the infinite relating paradoxes, I am very sure they don’t help me.

Sincerer yours, Geng

Dear Miguel, very happy for our frank discussions, thank you.

I also try to answer your four discussion points:

1, look at the widely accepted modern divergent proof of Harmonic Series

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）

According to your opinion, we have two kinds of infinite series in this proof:

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 + 1/2 + 1/2 + 1/2 + ．．．------>infinity ）                      （２’）

(=1+ 1 + 1 + 1 + 1 + ．．．------>infinity     ）                              （２’）

(=10000+ 10000 + 10000 + 10000 +．．．------>infinity   ）       （２’）

．．．

Both first the second kind of infinite series are infinite series with infinite items. According to your opinion, the first kind of infinite series is “higher” than the second kind of infinite series. Do they have same items (one to one correspondence) or first kind of infinite series has more items than the second kind of infinite series (not one to one correspondence) ------ first infinite series is more infinite than the second kind of infinite series because it has more items?

2, It doesn’t matter how you call the “elements” or “items” or whatever of those “Un--->0 things” in Harmonic Series, what we care about is: What are those “Un--->0 things” （or elements or items or infinitesimals …）in Harmonic Series? Can we produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite Harmonic Series by “brackets-placing rule"?

3, you are right that different people can have different cognitions of infinity (also according to different age, different orientation, different mood …). But when we are talking about mathematical operations, people (not matter what colors, what nations, what ages, what orientations, what moods,…) can only accept one of those cognitions ------ universal mathematical cognition.

4, our science history proves that any theories without the abilities of solving practical problems, they can only become “after dinner or tea time talks-----leisure for fun or Self-comforting”.

Thank you again dear Miguel.

Sincerer yours, Geng

Dear Miguel, thank you.

I will also answer to each point one by one:

1, According to your opinion, both first and the second kind of infinite series are infinite series with the same amount of infinite items in the above widely accepted modern divergent proof of Harmonic Series. So, do you mean we really can change the first kind of infinite decreasing series (Harmonic Series) with the property of Un--->0 into the second kind of infinite series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity? This is what I say the newly discovered “strict mathematically proven” Zeno’s Paradox: the “brackets-placing rule" to get 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite items in Harmonic Series corresponds to different runners with different speed in Zeno’s Paradox while the items in Harmonic Series corresponds to those steps of the tortoise in Zeno’s Paradox.

2, Harmonic Series is composed by infinite items, there are “Un--->0 things (items, elements…)” in Harmonic Series because of its infinite decreasing property. The “brackets placing rule” is operated on all the items in Harmonic Series including “those Un--->0 things” of cause.

3, we have many “thinking of humankind” (cognitive activities) such as cultural thinking, religious thinking, mathematical thinking …. When we are talking about mathematical operations, we can only accept one of those thinking ------ universal mathematical thinking. I think there are surely demarcations between and among cultural thinking, religious thinking, mathematical thinking …, what do you think on this point? Don’t you agree that we have universal mathematical thinking (the universal mathematical thinking may develop and evolve but it is there accompanying us human ever since)?

4, according to my teaching experiences, you are right that a good teacher is mastercraftsman. To different students with different ages (such as kinder garden pupils and university students) we may use different ways to satisfy them and reach our teaching effects-----the moon in the sky can be a cake, a gold coin, a paradise with immortals, an earth satellite,…. Magicians show us that they can create something out of nothing, we enjoy it but we know it is impossible.

Thank you again dear Miguel.

Sincerer yours, Geng

Our marvelous edifice of (so-called pure) mathematics is based on intuitions deriving from the evidence of 19th century physics (and elsewhere, I suppose).  This includes our inquiries into what it means to be infinite.  But--and I can hardly contain my glee at this--we may not be able to generate modern physical theory through our geometric intuitions, as developed over the past 200 years.  Quite possibly we have to get our hands dirty once again and delve into the world of the finite.  I guess--but I'm no expert on this--we have to develop tools that more closely link "infinite" with "large finite." ( I know I'm not expressing a radically new idea here.)

Dear Paul and Akira, I totally agree with you. It is true sometimes we meet difficult problems and can not help by some reasons.

The infinite related fundamental problems being disclosed are really difficult to be solved.

Looking back into our science history, a fact is discovered that many researchers have been focusing minds on how to create new “ideas, understandings and terms, formal languages” within the traditional finite—infinite theory system but not on understanding the nature of the fundamental defects disclosed by those “infinite related paradoxes”; so, the fundamental defects are still there unsolved or unavoidable, all the family members of “infinite related paradoxes” are still there unsolved or unavoidable challenging us human cognizing ability.

The “infinite related problems” have been really challenging us human at least since Zeno’s time. Something should be done on the foundation of the traditional finite—infinite theory system not matter how difficult the problems are whenever there is still human science in our human life, it is impossible to compromise or run away from those “infinite related paradoxes”.

Thank you for your frank opinion.

Sincerer yours, Geng

Dear Akira,

A saying goes like this: man proposes and god disposes.

I am deeply convinced that if our train of thought in this working field is on the right direction, no matter how much we have done, our human science will be improved more or less. 

We just try our best in our remaining life and that is enough!

My best regard to your, Geng

Geng and Akira,

This is now getting very important.  If everyone is under the assumption that there are undefined areas of math then we should just define them and look at the answers that it give us.  I am dealing with another problem that is related to this in some other groups.   There should be no undefined things in science even if they are defined incorrectly they need definition.  You can always work on corrections if there is a working model.

There can be no energy without mass.  So Einsteins zero rest mass just means that the definition was wrong.  Give this it means that all our problems with mass-less particles are wrong.  just because something has transferred all its energy to an atom or a mountain or a baseball or a particle of dust does not mean that it is not there any more. 

We have a very large problem with the uncertainty principle and quantum uncertainty by not understanding it or why it is even a part of the science.  We look at very large things and they act like we expect them to even as Newton and Einstein predicted. 

The problem is that we have these little bits of substence that in Quantum mechanics we ignore like it does not exist and we wonder why we can not predict what is happening.  

This type of ignoring the trees for the forest in front of you has been going on for longer than a century.  Einstein said that he just wanted to know what the Quanta was.  It is the part of light that we ignore because we think it is not there or has no mass.

This relates to the smallest of infinities and the biggest is related to limits that nature places on things which we (in our math) decide are infinite. 

I am talking about things like gravity.  In science and the theories of gravity there is no limit to its reach.  However in nature everything seems to have limits.  By placing no limits (which is a mathematical interpretation) there should be an infinite amount of energy that is lost due to the gravitational wave that is felt at an infinite distance in an infinite amount of directions.  This is or course is not the case in nature as nature employs reason.  As humans with the ability to ignore reason we sometimes choose to work with fails assumptions and make up the rest to mach the belief. 

Where is reason in the science that we want to believe is true.  Albert Einstein once said that the only thing that was infinite in the universe was human stupidity.  Maybe that is to harsh, but the idea is that we choose to ignore things and call others that point out the errors stupid for doing it.  

Geng and Akira you both point out the errors and I applaud you for that.

Dear George,

I think you made 2 important points:

1, human science is an alive thing existing along with our human evolution：metabolizing, developing to become “more scientific”. It is researchers duty to maintain our human science mansion.

2, the fundamental defects disclosed by the family members of Zeno’s Paradoxes in present infinite related science theory system make us unable to know “what those little bits of substances are”; so we just don’t know when to ignore and when to acknowledge their existence.  

You are right that something should be done not matter how difficult the project is.

My best regard to your, Geng

Dear Akira,

I am very sure that there is scientific honesty left in our science at any time in some people’s mind. The project is huge with many difficulties and something should be done as what you point out: theoretically and technically.

My best regard to your, Geng

Dear Akira, your are great in my mine because you are the one who always worries about the “health state” of our science and trying to do something.

It is reasonable that different people do different things arranged by “time and space”, so our science evolves and develops naturally and our world is colorful. It is science and it is the researchers’ life.

My best regard to your, Geng

Akira, you have presented some fallacious cases in physics dealing with infinite, infinitesimal, 0, limit theory, infinite related numbers,…, I think what you disclosed is a long being suspended syndrome of infinite related fundamental defects (confusions): what are infinite, infinitesimal, 0, limit theory, infinite related numbers,…? Anyone working in present traditional infinite related science branches (mathematics, physics, …) is sure to be confused with infinite, infinitesimal, 0, limit theory, infinite related numbers,… And some fallacious cases many be produced naturally by this syndrome.

Something should be done sooner or later to get rid of those infinite related fundamental defects and I am sure w it is now on the way.

Yours, Geng

If an entity can be measured (small or large) is it not finite? Is a human small or large? Compare the number of cells of a human with the number of stars in the visible universe. Is the universe finite or infinite? Proof?

Dear Mr. David C. Rine, your questions are very insightful and touch the foundation of present infinite related science theory system.

If we say “an entity can be measured (small or large)” it means this entity can be cognized with number forms in mathematics by finite related number forms (such as natural number 5, or 10000, …) or infinite related number forms (such as computing approaches with limit language for all kinds of Un--->0 number forms). As we know, there are no problems in the cognizing process with finite related number forms but there are still many problems in the cognizing process with infinite related number forms at least since Zeno’s time with his paradoxes because of the fundamental defects in present traditional “finite--infinite” related science theory system.

·      You are right, without a scientific foundation for “finite--infinite” theory system, people have been confusing with “small or large, finite or infinite, …?” when dealing things in our cognizing process (such as “Are the Un--->0 items in Harmonic Series small or large? How to treat them? ”; and the questions in your post: “Is a human small or large? Is the universe finite or infinite? ” …). And, no one now in the world can offer self-justification proofs in present traditional “finite--infinite” related science theory system to many “finite--infinite” related cognizing cases.

I am sure it is now on the way to get rid of those infinite related fundamental defects. This is a huge project.

Yours, Geng

Yes. However, 'infinite series have always been useful in engineering mathematics to come up with 'good enough or close enough' solutions to differential equations. In that case using close enough finite termination of an infinite series has always been useful.

You are right Mr. David C. Rine, 'good enough or close enough' solutions in engineering mathematics (applying mathematics) to physics is quite ok. But “applying mathematics” is only small branches of “theoretical mathematics (trunk)”, both “applying mathematics” and “theoretical mathematics” is needed in our science.

sometimes we complain that students are not taught “correct ‘finite—infinite’ related mathematic”, but now I fully understand that at least since Zeno’s time, everyone in present traditional “finite--infinite” related science theory system have to teach our students all the theories in our text book. After all, we have no choices and so do our students. Sometimes we are helpless to teach our students something we are not agree in the bottom of our heart but sometimes we teach our students “wrong mathematical things” unconsciously because we ourselves are within the defected present traditional “finite--infinite” related science theory system------ some fallacious cases are produced logically by many different reasons.

I think the thing really worries us scientists is: only a few people care how to avoid “purposely or unconsciously teaching our students wrong mathematical things” or how to solve the defects in present traditional “finite--infinite” related science theory system disclosed by the growing family members of “finite--infinite” related paradoxes.

Yours, Geng

Dear Miguel，very happy to hear from you again and thank you for your frank opinion.

1, you are right that I am basing all my argumentation and start my work in this field in the existence of paradoxes because I am deeply convinced that paradox is born by the defects in the related theory system.

2, the fact is Zeno’s Paradoxes have been defined and called more than 2500 years ago; they are still there accompanied by exactly “same and old defects” in our science today. It is not me who defined and called them paradoxes. I am sure wikipedia can tell you everything about Zeno’s Paradoxes in more detail.

3, I am very sorry that I disturbed many scientists’ peaceful life. But if what I said above is true, I will insist on my work in the rest of my life because those defects disclosed by the infinite related paradoxes should be removed from our science sooner or later-------by our children, grandchildren…

The facts presented above convinced me that I am on the right direction and really have done something already.

Thank you again Miguel. I really appreciate your opinions.

Yours, Geng