We really have had many different definitions for another half of the infinite (infinite small) in our present classical infinite related science system: infinite small, infinitesimals, non-number infinitesimal variables, actual infinitesimals,… in different periods of our infinite related science theory history such as “standard analysis” and “non-standard analysis”.

Why many people (including Cantor) think those “infinite small (infinitesimals)” should not be included in mathematical definition for “infinite”?

Actually infinity is not a real number so its opposite infinitesimal is also not real number.

Generally in arithmetic a/o = infinity that is not real and therefore division by zero is undefined. For mathematical impossibility of assigning a value to a/0 is criticised by George Berkeley in his criticism of infinitesimal calculus in The Analyst ("ghosts of departed quantities").

 Infinite small is as good as infinit big!

https://youtu.be/ZXKQFXlWvE0?t=1

Dear Dr. Mohammad Firoz Khan and Dr. Panagiotis Stefanides, thank you.

In fact, because of no mathematical definition for infinite in present classical infinite related science system, (1) it is impossible to solve The Second Mathematical Crisis in present classical infinite related science system because it is impossible to know what infinitesimals are mathematically; (2) more hidden infinite related paradoxes and troubles will be discovered in mathematic because it is impossible to know what “infinite things” being cognized are mathematically; (3) the “potential infinite” and “actual infinite” debates and troubles will keep challenging us human endlessly because both “potential infinite” and “actual infinite” are “non—mathematical things” but play important roles in present classical infinite related science system.

Best regards,

Geng

It can be included very well and has been included already in the work of Abraham Robinson. Consider T the theory of totally ordered fields, c a new constant and let N be the axioms "c > n" for all natural numbers n, written as 1 + 1 + ... + 1. It is trivial that every finite subset of T U N is consistent. Just take as model R with c bigger than all those n appearing in the finitely many axioms from N, which really occur in the finite subset. Then by the Compactness Theorem, T U N is consistent and has a model S. This model is a totally ordered field, contains an element "c" which is bigger than all natural numbers n, so 0 < 1/c < 1/n for all n. Otherwise said, 1/c is here strictly positive and infinitely small. S is a so called non-archimedean ordered field.

In fact the set O = { x in S | E n in N, | x | < n } is a valuation ring, called ring of limited numbers, and is the convex hull of Z in S, while M = {x in S | A n in N, | x | < 1/n } is the unique maximal ideal of O, and is the set of all infinitely small elements. Very interesting, both sets O and M are convex subsets of S and are centered in 0, but none of them is an interval of S.

Dr. Mihai Prunescu, thank you.

In fact, “infinite big” and “infinite small” seems to be included very well and has been included already in our classical limit theory (standard analysis) hundreds of years ago too as that of the work of Abraham Robinson; there have been already many different form languages with exactly the same nature in our history of classical infinite related science theory system based on the same classical “potential infinite” and “actual infinite” and some more may be “designed” sooner or later with exactly the same defects. Let’s see following example:

No one can tell what those “infinite things” of Un--->0 in Harmonic Series and dx --->0 in calculus are: “potential infinite” or “actual infinite”. Some people think Un--->0 in Harmonic Series and dx --->0 in calculus are both “actual infinite” while some think they are both “potential infinite”, still some think they can be both in different cases, ...

If we can not know what the “infinite things” exactly are, how can we behave in our practical mathematical operations?

 This question has been challenging and confusing us human for a long time.

Can standard analysis and non-standard analysis solve this question?

Can standard analysis and non-standard analysis solve the newly discovered Harmonic Series Paradox?

Best regards,

Geng

Dear Genk,

in the models of nonstandard analysis and more generally in any nonarchimedean  ordered field, the infinite elements and the infinitesimal elements are actual elements and not potential elements. The model has been constructed before. So whatever you express there is "actual".

I don't know  what do you mean by Harmonic Series Paradox.

If it is the next one here:

https://www.youtube.com/watch?v=x8-3OlB3G0M

then it is not really a paradox.

The line of length 1 + 1/2 + 1/3 + ... has no thickness, so we do not need a gram of ink to colour it. It has measure 0. If you imagine a line drown by ink as something of given thickness \epsilon, then the sequence of squares of areas 1 + 1/4 + 1/9 + 1/16 +... lying on the x axis becomes less thick then epsilon at a given step. So there is no paradox at all.

I have talk about Harmonic Series Paradox many times in RG, the following is what I mean by Harmonic Series Paradox-------it 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 science theory system. But, it is right with present modern limit theory and technology applied in this proof that we meet a “strict mathematical proven” modern version of Ancient Zeno’s Achilles--Turtle Race Paradox: the “brackets-placing rule" decided by limit theory in this proof corresponds to Achilles in Zeno’s Paradox and the infinite items in Harmonic Series corresponds to those steps of the tortoise in the Paradox. So, not matter how fast Achilles can run and how long the distance Achilles has run in “Achilles--Turtle Race”, there would be infinite Turtle steps awaiting for Achilles to chase and endless distance for him to cover, so it is of cause impossible for Achilles to catch up with the Turtle; while in this acknowledged modern divergent proof of Harmonic Series, not matter how big the number will be gained by the “brackets-placing rule" (such as Un’ >10000000000) and how many items in Harmonic Series are consumed in the number getting process by the “brackets-placing rule", there will still be infinite Un--->0 items in Harmonic Series awaiting for the “brackets-placing rule" to produce infinite items each bigger than any positive constants, so people can really produce infinite items each bigger than 1/2, or 100, or 1000000, or 1000000000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 1000000000000000), “strictly proving” that Harmonic Series is divergent. In so doing, the conclusion of “infinite numbers each bigger than any positive constants” can be produced from the Un--->0 items in Harmonic Series by brackets-placing rule and Harmonic Series is divergent" has been confirmed as a truth and a unimpeachable basic theory in our science (mathematics) while “the statement of Achilles will never catch up with the Turtle in the race” in Ancient Zeno’s Achilles--Turtle Race Paradox has been confirmed as a “strict mathematical proven” truth and a unimpeachable theorem------it would be Great Zeno’s Theorem but not Suspended Zeno’s Paradox! ?

Best regards,

Geng

The divergence has an easier proof, as this you speak about.

Let S = 1 + 1/2 + 1/3 + 1/4 + 1/5 + .... Suppose that S is a number.

Then S > 1/2 + 1/2 + 1/4 + 1/4 + 1/6 + 1/6 + ...., where we used that all terms of odd index are bigger then the terms of even index following immediately after them.

So S > 2/2 + 2/4 + 2/6 + 2/8 + 2/10 + ... = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... = S.

So if S is a number, then S > S. Contradiction.

So I did not need parantheses with exponentially growing distance between them! It is enough to look only at the odd terms and to group terms two by two (odd index with the next coming even index).

You will say that my proof is fallacious because I just shown that Sn = 1 + .... + 1/n is less then S2n = 1 + .... + 1/(2n), which is evident, but they can have the same limit. True remark,, but not quite complete. Because S2n - Sn = 1/(n+1) + .... + 1/(2n) > n x 1/(2n) = 1/2.

So if sequences S2n and Sn (which are sequences and not series) have the same limit S, which is a number, then S - S > 1/2, so 0 > 1/2. This time we have clearly a contradiction.

This time we used just one pair of parantheses!

Is there a paradox here? For me is just a diverging series.

Dear Dr. Anne Allaire , thank you.

It is good to hear your idea on 'infinitely small' and 'infinitely large' from musical view point.

Best regards,

Geng

Dear Dr. Mihai Prunescu, thank you.

I never said that Harmonic Series is convergent. In fact, either convergent or divergent is paradoxical in present infinite related science theory system.

Best regards,

Geng

Geng, of course you did not say this. I just thought that some other proofs of dívergence are maybe less paradoxical for you. For me the proof telling that S_{2^k} > k/2 is already very clear and not paradox at all. It is an inequality between well defined positive numbers, which can be immediately interpreted as divergence.

Dear Geng,

You are right at large. Infinity is a metaphysical concept. As Godel has pointed out that a consistent mathematical system has to import some thing from outside that it cannot prove. All this is Zeno's paradoxes. Suppose you have a machine that has unlimited memory and supply of energy placed in space. Programme it such that it goes on adding 1+1+1+........ such that it takes previous value for addition and removed all other values thus before addition of 1 it has only one value and all previous calculations are removed. Similarly suppose it dividing 1 by 2. 1/2, (1/2)/2, [{(1/2)2}/2}] and so on like the addition. If you believe in a universe which has a beginning and an end, then time will end but you will not get an infinite value or infinitesimal. If you believe in a universe that is forever the machine placed in space will go on calculating forever and would never stop because infinity or infinitesimal had not been achieved. Somewhere, I have discussed, that our best theories that use infinity or infinitesimal may be wrong at a certain point so that these values are returned.

The problem of infinity turns to be a true obsession that characterized the scientific and non-scientific Western World. Traditionally, the infinity was a term in the mystical and religious languages, which many thinkers use to shape their images of a pure spiritual reality. Thus, for instance, the boundless immensity of God is said to be a “sea” (a metaphorical sea), refers to a nonmaterial supreme reality, which human beings can attain only by transcending the nature and their earthly existence. In another theological system, that of negative theology presumes that God’s qualities or attributes are altogether different in nature, quality, or meaning to anything that human beings can know, presuppose or even dream; ergo, God is infinite and must be the opposite of the limited, bounded and finite human reality. God has the quality of being infinite, which everything that is created by God is limited. It follows that the idea of God was, is and always will be infinite Being that possesses infinite intelligence.

Later, the idea of infinity in the Early Modern Times is to be found at the basis of human being’s fascination with the vast, unlimited, and the suggestive Weltanschauung rather than the sharply defined terms or theories. This idea is bounded to another modern idea – namely, to the idea of an unbounded individuality which possesses unbounded power of will. Thus, the extension of the physical universe, as it has been demonstrated by the new revolutionary astronomy, seems to confirm the greatness of human soul.

Dear Anne Allaire , thank you very much for sharing with me your work。

I really like to know how you work with the concepts of 'infinitely small', 'infinitely large' and 'infinite' in musical field.  In fact, I am very sure that infinite exists everywhere in many fields of our human science with different carriers.

Best regards,

Geng

Dear Dr. Mihai Prunescu, thank you.

According to my studies, any convergent or divergent proofs with classical number theory (one of infinite carriers) and classical limit theory for Harmonic Series in standard analysis and its equivalent non-standard analysis are paradoxical in present infinite related science theory system basing on “potential infinite” and “actual infinite”. Because the infinite related number forms of the Un--->0 items in Harmonic Series can not be defined and treated mathematically.

Best regards,

Geng

Dear Dr. Mohammad Firoz Khan and Dr. Israel Bar-Yehuda Idalovichi, thank you.

I agree with you. In fact, the problem is at lest since Zeno’s time 2500 yeas ago, we human have been trapped deeply in an irresistible vortex of “confusing abstract concept of infinite (potential infinite) and its carriers (actual infinite without mathematical carriers)”-------such as the confusion of “abstract concept of fruit and its carriers of banana, apple, grape, …”. This fundamental defect disclosed by the old and new members of infinite related paradoxes families has been producing great ill effects on our quantitative cognizing activities to “infinite things”.

Best regards,

Geng

here is the answer

Dear Dr. Alejandro Alberto García De León Jiménez , thank you.

In fact, we have many definitions for the infinite related number forms of the Un--->0 in mathematics historically. But, the problem is so far no matter what definition is (standard analysis or its equivalent non-standard analysis or…), no one can tell “how many numbers each bigger than any positive constants (such as Un’ >1000000000000000)” can be produced from the Un--->0 items in Harmonic Series by the “brackets-placing rule" of present infinite related theory and skills.

It seems those historical definitions for the infinite related number forms of the Un--->0 are language games, not for solving mathematical problems-------that is the reason for “Why infinite small could not be included in mathematical definition for infinite in present classical infinite related science system?”.

Best regards

Geng

The conclusion of the pdf I wrote is that in R the real number,There isn't numbers "infinite small" that's because otherwise we have a contradiction with the continuous hypothesis with is a corolary from the supremum principle. moreover in mathematics we can study the concept of infinite sets even in advanced measure theory measure  numbers and in ordinals and cardinals theory we find uncountable examples of infinite numbers.

Dear Dr. Alejandro Alberto García De León Jiménez , thank you.

I agree with you that many people really don’t like to call the infinite related number forms of the Un--->0 "infinite small numbers" but like to call them “non-number number forms, variables, monads, …”. Well, it doesn’t matter how we call them (language games?), “the infinite related number forms of the Un--->0 (infinitesimals) in mathematics” are really there with infinities ------“how to define them mathematically” has been challenging us human at least since Zeno’s time 2500 years ago.

Best regards

Geng

If someone says: “Set A has limitless (boundless, endless) elements, so we say Set A is an infinite set; but if Infinite Set B has more elements than Infinite Set A, we say Infinite Set B is more infinite (more limitless, more boundless, more endless) than Infinite Set A”, what do we think?

If someone says “apple is fruit, banana is more (much) fruit than apple, pineapple is more more (much much) fruit than apple, pear is more more more (much much much) fruit than apple,… in our daily life, what do we think?

Can we really have infinite, more (much) infinite, more more (much much) infinite, more more more (much much much) infinite,… in our mathematics?

Cantor has done a lot in quantitative cognitions to “infinite things” within set theory, is it necessary and possible to improve and develop his work?

Dear Anne，

“Potential infinite” and “actual infinite” have been troubling us human since antiquity. Still, people are confusing on them all the time in most of our quantitative cognitions to “infinite things”-------endless debating on “potential infinite--actual infinite” and all kinds of infinite related paradoxes.

Regards,

Geng

Dear Geng,

From a formal point of view, the construction of what you call ‘infinitely big’ and ‘infinitely small’ is different.

Cantor's infinite cardinal hierarchy arises from his attempt to generalise the idea of equinumerosity from the finite to the infinite. The limit construction in mathematics comes from the need to define in a mathematically concise manner a concept that was used in physical theorising (17th-18th  century Wallis, Leibniz, Mercator, Kepler, ) where 1/\infty, dx, partial derivatives and tendencies of series were used without proper definitions. (Note that the limit concept handled both things ‘series diverging to infinity’ and a proper use of the ‘infinitessimal dx’ in derivatives.) Finally the construction of hyperreals which provides another meaning to the notion of an infinitesimal, only arose in the context of first order logics and non-standard models of field theory.  

One should also note the difference between the context in which a concept was proposed, and what it was later argued to represent. (Eg. hyperreals were not introduced to represent infinitesimals, although they are now thought to allow such representation and maybe much better than the limit construction. Similarly Cantor’s hierarchy of cardinalities was not introduced to represent the ‘infinitely big’, but was a side product of the attempt to formalise equinumerosity in purely set-theoretic terms).

There is, in principle, no good or bad in ‘infinities’ as there is no good or bad in any definition, as long as it is correctly and well defined. The question is rather, what you need them for (pragmatics of mathematics), what they allow you to understand (epistemology of mathematics) and whether you can do without it (ontology of mathematics). Your normative question of what should or should not be defined will have other meanings from these different perspectives and different answers, depending on what kind of stance you take on pragmatics, epistemology and/or ontology.

I hope this helps,

Best,

Eric

Dear Dr. Eric Raidl，

Thank you. But the problem is if there are really many different concepts of infinite with different meanings and definitions, 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.

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 Geng,

The limit concept is, I would say, the same in the definition of the convergence of a series and the definition of a derivative. But it acts differently - on a potentially infinite sum, with ever more terms vs. on a difference f(x-epsilon) - f(x) divided by the ever smaller term epsilon, which is a potentially infinite sequence of ratios.

I would teach people these concepts by motivating them starting with the problems that they were invented to answer and showing the underlying construction scheme. Eg. to understand what aleph_2 is you need to grasp the way Canor extended equinumerosity from the finite to the infinite (and maybe some properties of cardinal arithmetics). There is not much more to understand, unless you want to seriously do set-theory.

From a constructive point of view, and outside ZF, I would say that infinite cardinalities are ''potential'' (but in another sense than infinite limits) - the only similarity is that they arise based on a 'generation mechanism' which extends known concepts beyond the finite. However the 'generation mechanism' is different. From a set-theoretic perspective, I would say infinite cardinalities exist. What cardinalities exist depends on what you accept as further axioms to ZF. [Note that if you have ZF+AC+C you can presumably develop the whole of classical mathematics including limits, derivatives, hyperreals. So ZF +AC+C and maybe more is perhaps the 'overarching theory' you are trying to find....]

Somewhere you asked why there is no dual to 'infinitely big' in Cantor's set theory. It's just that there is no room for this. Eg. a singelton set cannot be ''divided'' in two ''equal'' sets. You would need to build a dual Cantor hierarchy, which'd be a kind of mirror image of the latter and see whether they behave well with respect to one another (multiplication, ''division''). I guess they won't. And I don't really see the need for this. Plus: it appears conceptually odd to me. But if you give it a try, tell me what comes out... 

Best,

Eric

Dear Dr. Eric Raidl, thank you.

“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. Vinoo Cameron, thank you for your view points.

Best,

Geng

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 following question scientifically and this is the very reason for many “suspended infinite related paradox families” in present classical infinite related mathematics:

Are “dx--->0 infinitesimal” in calculus (especially in applied mathematics) and “Un--->0 infinitesimal” in Harmonic Series the same things? 

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

Geng, You'd better start with presenting the definition of what you mean with "dx". I know at least three viewpoints.

1. The original viewpoint at the time of Newton and Leibniz looked at dx as the process of considering an ever smaller difference Delta (x) of successive values of a variable x. For centuries this process was mixed up with intuition and simply handled as a near-zero value. It gained lots of criticism, and the viewpoint (rather: the imagination of almost zero numbers) was abandoned during the 19th century. 

2. In differential geometry, differentials are considered as one-forms, informally, smoothly varying linear functions. I refer to the link for details.

3. In first-order logic one constructs non-standard extensions of the real line sharing all first-order properties of the "true reals". In addition, it possesses infinitesimals, which are positive members of the extension which are smaller than any positive "true reals". In particular these models are non-Archimedean. Several contributions of several authors (including myself) in several threads of yours already referred to the viewpoint of non-Archimedean fields.

In views 2 and 3 your question makes little or no sense. In view 1 your question is answered in a most elementary way: the successive members of the harmonic sequence (or of any other non-trivial sequence converging to 0) are an example of the process that dx stands for.

https://en.wikipedia.org/wiki/One-form

Dear Mr. Marcel Van de Vel, thank you!

It is true that some people really believe that infinite items each bigger than 1/2, or 100, or 1000000, or 100000000000000000000000,…can be produced from “Un--->0 infinitesimal” in Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as Un’ >10000000000000000000000000).

But, believe is one thing and practice is another thing: could you tell us how many items of “Un--->0 infinitesimal” in Harmonic Series you use to produce the first Un’ >10000000000000000000000000, how many items of “Un--->0 infinitesimal” in Harmonic Series you use to produce the second Un’ >10000000000000000000000000, how many items of “Un--->0 infinitesimal” in Harmonic Series you use to produce the third Un’ >10000000000000000000000000?

Geng,

You do not mention whether my last contribution answered your previous question on "dx" and the Harmonic series 1 + 1/2 + 1/3 + .... . Did it?

I vaguely remember having answered your last question  a few years ago in some other thread of yours on the same subject. Rather than repeating it, I suggest you take a look at (for instance) the Wikipedia paper of the link, which proves that the sum of the first 2k members of the series is larger than 1 + k/2.

E.g., the sum of the first 2 to the k =20000000000000000000000000th terms is larger than the number 10000000000000000000000000 which you propose as a challenge. If you are a mathematician, you will surely understand the elementary proof of the Wikipedia paper. In this way your last question is finally answered; I will not return to it again.

Most cultures have a wisdom stating that "many small things can make something big". It also applies to mathematics.

https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)

Dear Mr. Marcel Van de Vel, thank you very much for your frank ideas!

Do you really think that it is so easy for you to get the first, the second and the third Un’ >10000000000000000000000000 from “Un--->0 infinitesimal” in Harmonic Series.

But the fact is: can you really prove it but not just guess it-------the sum of the first 2 to the k =20000000000000000000000000th terms is larger than the number 10000000000000000000000000.

It is good that you are always frank. Thank you again and many people are really interested in your proofs for the first three 10000000000000000000000000 from “Un--->0 infinitesimal” in Harmonic Series.

Best regards

Geng

Geng,

As a mathematician you should know the difference between a mathematical proof and a "guess". And, by the way, it is easy to get the second, third, etc. portion of the Harmonic series that sums to  more than10000000000000000000000000 (= 1025, I think): the first portion consists of the first (2 to the power 2*1025 ) terms, the second portion consists of the next terms up to the (2 to the power 4*1025 )th term, the third involves the next terms up to the  (2 to the power 6*1025 )th term, etc.

To effectively build these sums takes more time than the universe exists.  Fortunately, mathematics is about what is logically possible, not about what is physically possible. That's the essential feature of a mathematical proof. Euclid proved that there isn't a largest prime number. Can you give me the 10000000000000000000000000th prime number, dear Geng? It does exist! Or do you believe Euclid is wrong? 

As to people interested in a proof of the above statements on the harmonic series I refer to the Wikipedia paper; it is not my proof. The statements about the second and third portion can easily be deduced and even extended to the fourth, fifth, ..., portion with a similarly large sum.

An infinitesimal is a numeric function or sequence that tends to zero.

Infinitely large - a numerical function or sequence that tends to infinity of a certain sign

The concept of "infinitesimal" was discussed even in ancient times in connection with the concept of indivisible atoms, but it did not enter classical mathematics. Again, it revived with the appearance in the sixteenth century of the "method of the indivisible" - the partition of the figure under study into infinitesimally small sections.

In the XVII century there was an algebraization of the calculus of infinitesimal. They began to be defined as numerical values that are smaller than any finite (positive) value and yet are not equal to zero. The art of analysis consisted in the construction of a relation containing infinitesimal (differentials), and then - in its integration.

The mathematicians of the old school subjected the concept of infinitesimal to sharp criticism. Michel Rolle wrote that the new calculus is a "set of brilliant mistakes"; Voltaire pointedly remarked that this calculus is the art of calculating and accurately measuring things whose existence can not be proved. Even Huygens admitted that he did not understand the meaning of differentials of higher orders.

Disputes in the Paris Academy of Sciences on the substantiation of the analysis have become so scandalous that the Academy once even banned its members from speaking on this topic (mostly about Roll and  Varignon). In 1706, Roll publicly withdrew his objections, but the discussions continued.

https://en.wikipedia.org/wiki/Infinity

https://www.youtube.com/watch?v=9xmJUSJ4EwA

Dear Mr. Marcel Van de Vel, thank you very much!

1, I am very sorry to say that what you said above “And, by the way, it is easy to get the second, third, etc. portion of the Harmonic series that sums to  more than10000000000000000000000000 (= 1025, I think): the first portion consists of the first (2 to the power 2*1025 ) terms, the second portion consists of the next terms up to the (2 to the power 4*1025 )th term, the third involves the next terms up to the  (2 to the power 6*1025 )th term, etc.” is really nothing to do with mathematical proof. Is it only your own guess or imagination?

2, if you believe that you really can prove the first, the second and the third Un’ >10000000000000000000000000 from “Un--->0 infinitesimal” in Harmonic Series, why not get the helps from computer but not by your own labors?

3, no one have seen the Wikipedia papers for the proofs about the first, the second, the third and even extended to the fourth, fifth, ... Un’ >10000000000000000000000000 from “Un--->0 infinitesimal” in Harmonic Series.

Best regards

Geng

Dear Mr. P.F. Zabrodskii , thank you very much for the information!

It is true that the history of infinite has been long.

Paradox is the touchstone for the defects in the very science area. My studies proved that the suspended "infinitesimal related paradoxes" in mathematics is a strong “metabolism signal for the basic theory” which tells straightforward to scientists that something must be done to solve those fundamental defects disclosed by the paradoxes and get rid of the “paradoxes generating” fundamental defects.

------  “New infinite idea", "new mathematical carrier of abstract infinite law", "infinite related new number system", "new mathematical carrier related number form", "new infinite number form related new operation theory" is the foundation of new mathematical analysis..

Best regards

Geng

The statements about portions of the harmonic sequence summing up to be >= 1025 are an elementary consequence of the cited formula:

(*)     (sum for n from 1 to 2k of 1/n) >= 1 + k/2.

Most mathematicians would feel insulted if someone explained a simple argument to them.

For computer assistance supporting the sum formula (*), consult the link or use Maple or Mathematica. They can't handle a sum of 2 to the 1025th terms. That's what a mathematical proof is for.

Geng, did you find the 1025 th prime number? Of course not. Do you conclude that it does not exist? Or that Euclid just guessed his result? You have avoided all my current questions and several others before. Your failure to distinguish a mathematical proof from a "guess" obstructs any further discussion of the subject.

I quit.

http://www.dcode.fr/hamonic-number

Dear Mr. Marcel Van de Vel, thank you very much for your frank ideas!

If it is impossible to get just first three Un’ >10000000000000000000000000 from “Un--->0 infinitesimal” in Harmonic Series, how you are so sure to get infinite Un’ >10000000000000000000000000 from “Un--->0 infinitesimal” in Harmonic Series?

Best regards

Geng

You should read more carefully, Geng.

For each integer k>0 the k-th portion of the harmonic sequence, consisting of the terms numbered from

            (2 to the power 2k-1*1025)    to    (2 to the power 2k* 1025)

sum up to be >= 1025. To be sure, this is finitary mathematics, it does not appeal to anything infinte, even the ancient Greeks could read and accept the argument, so don't blame the "infinite related science system" for this.

You are plainly wrong, and obsessive in this matter. As you do not see the difference between a mathematical proof and a guess, there is no basis for a further discussion.

For some reason, I can not activate the "edit" button in my previous contribution to correct a typo.

Correction: the k-th portion of the harmonic sequence (1/n), n = 1,2,3,... runs for n

        from (2 to the power 2(k-1)*1025) to (2 to the power 2k* 1025).

Dear Mr. Marcel Van de Vel, thank you very much for your frank ideas!

But I am very sorry to tell you that you are the only one has been known in this world so sure to get the first three Un’ >10000000000000000000000000 from “Un--->0 infinitesimal” in Harmonic Series by above way.

And what is more------ the only one has been known in this world so sure to get infinite Un’ >10000000000000000000000000 from “Un--->0 infinitesimal” in Harmonic Series by above way.

Best regards

Geng

To be the only person in the world with such knowledge would be a great honor if it were not assigned by someone who hardly reads what I wrote in this discussion.

Didn't you notice my words "finitary mathematics", Geng?

Didn't you notice how I carefully avoid any reference to infinity or infinitesimals?

Didn't you notice that I communicate common knowledge from the public domain?

Mathematical definitions of the mathematical objects form a consistent system of notions. The notion of "infinity by  Geng Ouyang" is different, i.e. it is an element of another system of notions and therefore it is indisputable. Neither as a mathematical object (as a being outside the system), nor as an (un)acceptable element of a new theory (since everyone has right to build his/her system of notions).

Regards, Joachim

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

1，On the Quantitative Cognitions to “Infinite Things” (I)

2，On the Quantitative Cognitions to “Infinite Things” (II)

3，On the Quantitative Cognitions to “Infinite Things” (III)

4 On the Quantitative Cognitions to “Infinite Things” (IV)

5 On the Quantitative Cognitions to “Infinite Things” (V)

Dear colleagues,

The new ideas in The Late Coming Actual Infinite Theory System:

In the new infinite theory system basing on scientific concepts of "abstract infinite and the carriers of abstract infinite", we abandon the concepts of "potential infinite and actual infinite", replace the concept of "potential infinite" by the concept of "abstract infinite", and replace the concept of "actual infinite" by the concept of "carrier of infinite". From now on, human beings are no longer entangled in "potential infinite -- actual infinite", but can spare no effort to develop "infinite carrier theory", and develop comprehensive and scientific cognition of various contents related to "mathematical carrier of abstract infinite concept", the contents of "abstract infinite concept (abstract scientific thing) - mathematical carrier of abstract infinite concept (the carrier of abstract scientific thing)" have been discussed and introduced.

We introduce "Infinite Mathematical Carrier Gene (Abstract Concepts Carrier Gene)" concept to describe the basic construction and properties of infinite mathematical carriers ------- it was the born "carrier gene" that determines each "infinite mathematical carrier" has its unique characteristics, unique existence conditions and the unique form of expression, guarantees us conducting down-to-earth scientific cognitive activities (of course including quantitative cognition) to infinite mathematical carriers.

A new concept of Infinite Mathematical Carrier Measure is introduced basing on the concept of "Infinite Mathematical Carrier Gene". The new concept was created specifically for those smallest or biggest quantitative cognitive units that retain the natures of "infinite mathematical carriers”, so that we can conduct the scientific and efficient qualitative and quantitative cognizing operations to different "infinite mathematical carriers” from different perspectives---------- smallest or biggest velocity, smallest or biggest rational number, the whole quantity of elements in Natural Number Set, the whole quantity of elements in Real Number Set, different quantity of elements in different infinite sets, ...; the Infinite Mathematical Carrier Measure of elements in Natural Number Set is smaller than that of in Rational Number Set or Real Number Set because the "Natural Number Carrier Gene" determines the Natural Number Set contains only elements with "the "Natural Number Carrier Gene" but none of the other elements (such as the rational numbers or real numbers) with other “carrier gene", namely "the infinite carrier gene of rational number or real number is greater than the infinite carrier gene of natural number" determines that the infinite carrier measure of elements in Rational Number Set or Real Number Set is sure be bigger than the infinite carrier measure of elements in Natural Number Set;

Our studies have proved that the "tangible, visible, and reachable actual infinite" that people have been talking about and using for thousands of years basing on their senses and without any justification, should have been a set of "New Infinite Carrier Theory System (The Late Coming Actual Infinite Theory System?)" that includes following new mathematical things which are closely related to the concept of "infinite": the new number form, the new Number Spectrum, the new definition for infinite set, the new definition for the elements of infinite set, the new “set – elements” relationship, the new Set Spectrum, the new “Semi-Archimedean Character”, the new “number character”, the new “set character”, the new Infinite Mathematical Carrier, the new Infinite Mathematical Carrier Gene, the new Infinite Mathematical Carrier Measure, the new limit theory and its operation, the new one to one correspondence theory and its operation, ...". The new infinite theory system (especially the newly developed "infinite mathematical carrier theory") lays a solid and scientific foundation for the second generation of set theory and the fourth generation of mathematical analysis for human to scientifically carry out the qualitative and quantitative cognitions of infinite mathematical things. We should cherish and inherit the infinite related wealth of knowledge gained by our predecessors, but we should also correct mistakes produced by the unscientific classical infinite theory system and open up new fields.

Sincerely yours,

Geng