The fact is: theoretically, the present science theory system has really proved the divergence of Harmonic Series; but practically, Harmonic Series is impossible to be divergent because it is impossible to change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 10000000000) by the “brackets-placing rule" -------in present science theory system, it is impossible for us to prove how many items in Harmonic Series are consumed for the first 10000000000 by the “brackets-placing rule" and how many for the second 10000000000. If the “brackets-placing rule" is unable to produce just only two 10000000000, how can infinite items of 10000000000 come and how can Harmonic Series be divergent? We call such a newly discovered logical contradiction “Paradox of Harmonic Series”.
The acknowledged modern divergent proof of Harmonic Series is one of the family members of ancient Zeno’s Achilles--Turtle Race Paradox. Of cause, the family of Zeno’s Paradox was not born just for telling us “Achilles is really unable to catch up with the Turtle in the race forever” or “infinite numbers each bigger than any positive constant can really be produced from Harmonic Series by brackets-placing rule”, but for disclosing fundamental defects in present classical infinite related science system: the defects in philosophy of mathematics and philosophy of science, in infinite conceptual system, in limit theory, in the “infinite nature carriers”, in the scientificity cognition of mathematical knowledge,….
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 10000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 10000000000), “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! ?
Dear Professor Stefan Gruner , thank you very much for your view points.
Forgetting about that “pseudo Zeno's paradox”, how do you think of the genuine mathematical Harmonic Series Paradox?
Best regards,
Geng
Dear Professor Stefan Gruner , thank you.
I think at less since Zeon’s time, people have been paying too much attention to the forms but too less to the ontology so we can not really understand what infinite is and debating endlessly on “form and ontology”------potential infinite and actual infinite. So, 2500 years ago we had Zeno's Paradox and now we have Harmonic Series Paradox. There will sure be some other members of Zeno's Paradox families now and then if those defects disclosed by the confusing and trouble making “potential infinite and actual infinite” are not solved.
In present classical infinite science system, the members of Zeno's Paradox families disclosed that it is unavoidable that we mix (jump between) “potential infinite” and “actual infinite” whenever we cognize any infinite relating things.
Kind regards,
Geng
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