The following proof (given by Oresme in about 1360), very elementary but important, can be found in many current higher mathematical books written in all kinds of languages.
1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．． （1）
=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．． （２）
>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．． （3）
=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity （4）
Each operation in this proof is really unassailable within present 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[1-3]: 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’ >100000000000000000) 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 100000000000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 100000000000000000), “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! ?
There are two reasonable “standard” or “nonstandard” operations for the items in Harmonic Series in present science system:
（1） During the whole process in dealing with infinite substances (infinitesimals) in calculations, no one dare to say “let them be zero or get the limit or get the standard number”. So, the infinitesimals in the calculating operations would never be too small to be out of the calculation formulas and the calculations would be carried our forever. This situation has been existing in mathematics since antiquity------ those items of Un--->0 never be 0 all the time and Harmonic Series is divergent, we can produce infinite number items each bigger than1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite Un--->0 items in Harmonic Series and 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. Here we have one of the modern versions of ancient Zeno’s Paradox.
（2） During the process in dealing with infinite substances (infinitesimals) in calculations, someone suddenly cries “let them be zero or get the limit or get the standard number”. So, all in a sudden all the infinitesimals in the calculations become too small to stay inside the calculations, they should disappear from (be out of)any calculation formulas immediately. This situation has been existing in mathematics since antiquity-------those items of Un--->0 must be 0 from sometime and Harmonic Series is not divergent, so we cannot produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000or…from infinite Un--->0 items in Harmonic Series. But if it is convergent, another paradox appears and more troubles happen.
How and what can we do with intuitionism philosophy of mathematics to solve this paradox?
How and what can we do with formalism philosophy of mathematics to solve this paradox?
How and what can we do with logicism philosophy of mathematics to solve this paradox?
…
If there are the fundamental defects in certain science branch disclosed by paradoxes, it would be the right time for the “philosophy of that science branch” to play a role. Not matter how beautiful and how complicate a “philosophy of mathematics” is, it should always remember its own form and one of the important missions: not only mathematics and not only philosophy, but the very integrating basic stone of mathematics and philosophy guaranteeing the clear nature and peaceful existence of mathematics without being disturbed by mathematical paradoxes.
The infinite related paradoxes have been troubling us for such a long time------the fundamental defects are not only mathematics and not only philosophy. Is it the faults of our not having been done enough on “philosophy of mathematics”?
What is “philosophy of mathematics” for? Do we need both theoretical and applied “philosophy of mathematics”?
Is it too much requirements for “philosophy of mathematics”?
As far as I understand both philosophy of mathematics and your question, I can say that the philosophy of mathematics as a scientific domain does not answer to questions put in this way.
For example, there is a very old discussion concerning Zeno's Paradoxy of Achiles and the Tortoise. There are some philosophers defending the position that the fact that the series "sum (1/10)^n" converges, completely solves the paradoxy. There are other philosophers considering that this does not solve the paradoxy at all, and the problem is quite different, touching questions about predictibility and determinism in nature, or many other points of view. The discussion is going on, and will be reopened at least once per century, with newer and stronger arguments on both sides. What arguments, this depends of the general progress in science and philosophy.
On your special question: one can change the for-signs of the armonic series such that it oscilates over any compact [liminf a, limsup a] or such that the resulting series converges. Moreover, if the chosen limit is a computable real number, then the function giving the corresponding for-signs is also computable. For me this is a completely understandable mathematical statement. Do we really need philosophy here? [We should have needed it if the statement had contained something about the real world, like Zeno's Paradoxy... But do we need philosophy for this statement?...]
Dear Mihai and Ramon，
I think both of you heat the points from different aspects on infinite related paradox and you high light the very problems. Thank you very much!
1, if the discussions on Zeno's Paradox of Achiles and the Tortoise (a “scientific fact” for us to cognize) will be reopened at most once per century, it means that some fundamental things in our science system are still “unscientific” as far as the time being. Only the philosophy of mathematics can help us to know well those fundamental things in our mathematics: What is infinite (potential infinite or actual infinite or …)? What is (infinite related number forms: infinitesimal, infinite great, …)? What is the calculation operation for infinite related number forms (limit theory or proximate theory or …)?
Zeno's Paradox of Achiles and the Tortoise is only an inciter, a very powerful and stubborn inciter, challenging us human for thousands of years. I think Zeno's Paradox of Achiles and the Tortoise is a very typical case with more philosophy taste for the philosophy of mathematics.
2500-year passes in a flash but Zeno's Paradox is still there. Where is philosophy of mathematics?
2, The newly discovered Harmonic Series Paradox with more mathematics taste for the philosophy of mathematics discloses exactly the same fundamental defects in our science system (mathematics) as Zeno's Paradox does: What is infinite (potential infinite or actual infinite or …)?What is infinite related number forms (infinitesimal, infinite great, …)? What is the calculation operation for infinite related number forms (limit theory or proximate theory or …)?
The Harmonic Series Paradox is right there in our mathematics. The very thing it worries me much is this family member of Zeno's Paradox of Achiles and the Tortoise may challenge us human for other thousands of years?
Science paradoxes are purposely created for the fundamental defects in our science system.
Sincerely yours, Geng
You are right dear Ramon, we may have many different ideas of infinite (cultural, religion, scientific,…), for thousands of years: “mathematicians work wonders with this concept in all its varied forms, they do not have a monopoly on it”. “The infinite” is really something special for the philosophy of mathematics, but science is our human’s, we are sure to work out and own a scientific one. This is what the philosophy of mathematics for.
Sincerely yours, Geng
Driven and powered by the newly discovered infinite related Harmonic Series Paradox, we have very clear target which is different from those of previous “just talks and writings”------hoping to solve some of the newly found infinite related fundamental defects in our mathematics (science, philosophy) through the discussions here.
Facing the newly discovered family member of 2500-year old Zeon’s Paradox------Harmonic Series Paradox, something should be done.
We have now more than a dozen of “philosophy of mathematics”; they are not being for “after dinner or tea time talks” but for the healthy foundation of mathematics.
Dear Mr. George Stoica,
I totally agree with your idea so I said: in front of Zeon’s Paradox, although we have now more than a dozen of “philosophy of mathematics”, they are shall not being for “after dinner or tea time talks” but for the healthy foundation of mathematics.
The Theory of Relativity relating to “time--space” idea in the philosophy of physics (“absolute space--relative space” and “absolute time--relative time”) successfully solved the defects in the foundation of classical physics disclosed by the “Time—Space Paradox” in late 19th and dispel “A piece of dark clouds over the physics”.
It is the right time for philosophy of mathematics to do something like philosophy of physics now, we are on the way!
Sincerely yours, Geng
Dear Mr. Vitaly Voloshin，
Why you do not think that philosophy of mathematics can solve specific mathematical problems?
Is following a specific mathematical problem: There have been so many “answers” to what “potential infinite” and “actual infinite” are in present science system (philosophy of mathematics), but these answers actually answer nothing at all-------what are the differences between them and how to have quantitative cognizing operations to the “infinite thing” in from of us such as those “Un--->0 things” in Harmonic Series?
Sincerely yours, Geng
I'm quite sure that they can give nice answers to any unsolved problems we have to deal with, including universe or quantum. I'm really not a specialist for it but I feel that's a really good field of research to find an issue to our fundamental questions...
the best to you all
Dear Mr. Vitaly Voloshin，
The “2500—year huge black cloud of infinite related paradoxes over mathematics sky” can only be dispelled by philosophy of mathematics.
There is to be a revolution in philosophy of mathematics and philosophy of science.
Sincerely yours, Geng
Dear Mr. Thierry W. LEVI-FAICT，
Thank you ever so much for your encouragements.
Sincerely yours, Geng
I hope, so many “infinite theories” introduced in present science system ever since are not being “irrelevant chatting things after dinner and tea time”. I sincerely wish, armed with those infinite theories introduced in present science system, someone in the world now can solve the stubborn defects disclosed by newly discovered infinite related Harmonic Series Paradox.
The five typical defects( the “time--space” absence defects in philosophy of mathematics, the defects on infinite conceptual system, the defects in limit theory, the defects on the “infinite nature carriers”, the cognitive defects on the scientificity of mathematical knowledge) disclosed by the members of Zeno’s Paradoxes Family have been there challenging us for more than 2500 years, unsolved and unavoidable. This huge black cloud of infinite related paradoxes over mathematics sky” can only be dispelled with the help of philosophy of mathematics.
In present classical “multi--infinite--definition” related science, when facing “infinite something”, one at lest have two troubles:
1, “potential infinite” or “actual infinite” or “super infinite” or “sub infinite” or “secondary infinite” or “lower infinite” or…?
2, how much “endless, limitless and infinite” it would be?
So, just follow someone’s ideas doing something in present classical infinite related science:
1, we are wrong in doing something although we didn’t know why, just do (such as all the family members of “strict mathematical proven” Suspended Zeno’s Paradox);
2, we are right in doing something although we didn’t know why, just do (such as most of the successful work in analysis and set theory).
Dear colleagues, anyone knows how to solve following problems:
1, According to the definition, the original Even Number Set and Odd Number Set is each half of the Natural Number Set (otherwise it can not be called Even Number Set and Odd Number Set). People have been using two deformed Natural Number Set of (1x2, 2x2, 3x2,...,nx2,...) and (1x2+1, 2x2+1, 3x2+1,...,nx2+1,...) and just used the components of even numbers and odd numbers (natural numbers, but not the whole thing of“2,4,6,8…2n” and “1,3,5,7,9…nx2+1”) to map natural numbers, of cause they map very well because it is N ----> N itself but not 2n ----> N and nx2+1----> N.
What are mapping to natural numbers: the even numbers and odd numbers or the components of even numbers and odd numbers?!
2,The elements of tiny portion of rational numbers from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) map and use up (bijective) all the numbers in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so infinite rational numbers (at least 2,3,4,5,6,…n,…) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set”------- Rational Number Set and Natural Number Set are not bijective at all.
What decides the same cardinality of two sets?
In present classical infinite science system, people have been paying too much attention to the forms but too less to the ontology at least since Zeon’s time 2500 years ago.
Regards,
Geng
Five of my published papers have been up loaded onto RG to answer such questions:
1，On the Quantitative Cognitions to “Infinite Things” (I)
https://www.researchgate.net/publication/295912318_On_the_Quantitative_Cognitions_to_Infinite_Things_I
2，On the Quantitative Cognitions to “Infinite Things” (II)
https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II
3，On the Quantitative Cognitions to “Infinite Things” (III)
https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III
4 On the Quantitative Cognitions to “Infinite Things” (IV)
https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV
5 On the Quantitative Cognitions to “Infinite Things” (V)
https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V
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