Of cause in present infinite related mathematics, Russell did not go wrong in Russell’s Paradox at all. Because the exactly same mechanism of Russell’s Paradox (the “prodution line” of looking for something belongs to an infinite set but is impossible to be found inside this infinite set) can not only just be used to produce suspended Russell's Paradox but also produce fundamental theorems of modern “Power Set Theorem” and “Uncountability of Real Number Set”.
Within present classical “potential infinite--actual infinite” based set theory, no one dares to claim Cantor did anything wrong in his works on the fundamental theorems of modern “Power Set Theorem” and “Uncountability of Real Number Set” with exactly the same mechnisim of Russell’s Paradox.
Studies proved that the members in Zeno's Paradox Family and Russell's Paradox Family form a “infinite paradox’ symptom complex” form a “infinite paradox’ symptom complex”.
Thank you dear Dr. Manoj Sithara
Something more than a “Barber":
1, The exactly same mechnisim of Russell’s Paradox (the “prodution line” of looking for something belongs to an infinite set but is impossible to be found inside this infinite set) can not only just be used to produce suspended Russell's Paradox but also produce fundamental theorems of modern “Power Set Theorem” and “Uncountability of Real Number Set”.
2, Nothing and no one is able to stop Cantor in his works on the fundamental theorems of modern “Power Set Theorem” and “Uncountability of Real Number Set” with exactly the same mechnisim of Russell’s Paradox (the “prodution line” of looking for something belongs to an infinite set but is impossible to be found inside this infinite set).
Yous,
Geng Ouyang
Thank you dear Dr. Manoj Sithara
Something more than a “Barber":
1, The exactly same mechnisim of Russell’s Paradox (the “prodution line” of looking for something belongs to an infinite set but is impossible to be found inside this infinite set) can not only just be used to produce suspended Russell's Paradox but also produce fundamental theorems of modern “Power Set Theorem” and “Uncountability of Real Number Set”.
2, Nothing and no one is able to stop Cantor in his works on the fundamental theorems of modern “Power Set Theorem” and “Uncountability of Real Number Set” with exactly the same mechnisim of Russell’s Paradox (the “prodution line” of looking for something belongs to an infinite set but is impossible to be found inside this infinite set).
Yous,
Geng Ouyang
Thank you dear Dr. Aleksandar Jovanovic
Of cause all the paradoxes constructer (including Russell, even you and me) in present classical set theory and analysis did not do anything wrong because people can only do according to the theory system.
Our science history has proved that the existence of paradoxes in certain area of human science is not only the touchstone for the defects in the very science area (such as the suspended paradoxes in set theory) but also the strong “metabolism signal for the basic theory”, and the “‘infinite paradox’ symptom complex” tells in its special way to scientists clearly that something must be done to solve those fundamental defects disclosed by the paradoxes and get rid of the “paradoxes generating fundamental factors” in that very science area.
Yous,
Geng Ouyang
As, I have argued in the case of objective approach (p. 373) my paper: https://www.researchgate.net/publication/286623797_Quality_of_Urban_Environment_A_Critical_Review_of_Approaches_and_Methodologies pointed out “Kurt Friedrich Gödel dealt a devastating blow to empirical science around the first quarter of the 20th century. Gödel’s incompleteness theorems say that any system that is complex enough to express in mathematics cannot prove by itself that everything it says is true. It will always rely on something outside the system that one has to assume is true, but cannot prove. In any consistent formal logical system of statements, there will always be a statement that cannot be proved (“a creative intuition” in words of Karl Popper), making the entire system incomplete.
“Gödel’s incompleteness theorems imply that all consistent conceptual systems, including logic and mathematics and, by extension, sciences are incomplete.”
So, Russell’s Paradox is based on axiomatic mathematical logic. Since, axioms cannot be questioned, so arithmetic and by extension set theory also cannot be questioned. As Gödel's theorems of mathematical logic show the inherent limitations of formal axiomatic system, Hilbert's program to find a complete, finite and consistent set of axioms for all mathematics is impossible. https://en.wikipedia.org/wiki/Hilbert%27s_program
It is why that by another axiomatic set theory, ZFC after Zermelo, Franekel and Skolem resolves Russell's and many other similar paradoxes by adopting axiom of choice. The theory reached resolution by defining a set of axioms in which it is not necessarily the case that there is a set of objects satisfying some given property, unlike naive set theory in which any property defines a set of objects satisfying it.
The axiom of choice is an axiom in set theory with wide-reaching and sometimes counterintuitive consequences. It states that for any collection of sets, one can construct a new set containing an element from each set in the original collection. In other words, one can choose an element from each set in the collection.
It must be noted that Developments in logic in the 20 century established that the axiom of choice was independent of the commonly accepted axioms of set theory due to Zermelo and Fraenkel or by extension that used by Russell’s. Again, Gödel is proved right, i.e., Zermelo-Fraenkel set theory with the axiom of choice (ZFC) and Zermelo-Fraenkel set theory based on classical mathematical knowledge that attempts to solve Russell’s paradox without the axiom of choice (ZFC) are both logically consistent theories. Nevertheless, most mathematicians generally accept the axiom of choice as true and use it when necessary.
https://www.google.co.in/imgres?imgurl=https://i.ytimg.com/vi/WIrdyu9WquQ/hqdefault.jpg&imgrefurl=https://www.youtube.com/watch?v%3DWIrdyu9WquQ&h=360&w=480&tbnid=Xupcd6ds5FFAoM:&tbnh=158&tbnw=211&usg=__jZ2PYbjZuPvzwz7SHcTQzdshVIQ%3D&vet=10ahUKEwjK-6i6hcHYAhXGpo8KHXzKARQQ9QEILDAA..i&docid=6bMcNaHo9GyhAM&sa=X&ved=0ahUKEwjK-6i6hcHYAhXGpo8KHXzKARQQ9QEILDAA
Thank you very much dear Dr. Mohammad Firoz Khan,
Yes. We have to accept the fact: nothing and no one is able to stop the “production line of looking for something belongs to an infinite set but is impossible to be found inside this infinite set” in present infinite related science and mathematics-------applying exactly the same mechanism of Russell’s Paradox (with the “routine production line operation” of looking for something belongs to an infinite set but is impossible to be found inside this infinite set), every one can prove two important results of modern“power set theorem” and “Real number set is uncontable”.
Yous,
Geng Ouyang
Russell was wrong by using classical logic.See for example
https://link.springer.com/chapter/10.1007/978-94-007-4438-7_17
Arruda, A.I., and D. Batens. 1982. Russell’s set and the universal set in paraconsistent set theory. Logique et Analyse 25: 121–133.
http://www.cle.unicamp.br/prof/coniglio/Carnielli-Coniglio-ParST.pdf
Thank you for your ideas dear Dr. Marcel Van de Vel ·,
People have been talking about members in Zeno’s Paradox Family for more than 2500 years, but I have only talked about members in Zeno's Paradox Family and Russell's Paradox Family (my studies proved they form a “infinite paradox’ symptom complex”) for more than 40 years----please don’t worry about that.
The problem is the abcence of basic theory leads to arbitrary operations and formal languages: an exactly same pipeline operation and formal language can be used to produce totally different products:
1, In analysis: on the one hand, any one can use “the ‘potential infinite--actual infinite’ confusing formal language and production line” to construct all kinds of infinite related paradoxes; on the other hand, one can also use exactly the very same “formal language and production line” to construct all kinds of infinite related theorems. The typical example is: Zeno’s construction (proof) of “Achilles Can Never Chase Up Turtle Paradox” and its modern version of “creating infinite items each greater than 1/2 or 100 or 1000000000000000 or 1000000000000000000000000000000 or ... from the Un--->0 Harmonic Series and turn the Harmonic Series into a ‘Vn ---> any positive constants’ infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000)”.
2, In set theory: the “one-to-one coresponding operation and formal language” can not only be used for the result of “Rational Number Set is equal to Natural Number Set” but also for the result of “Rational Number Set is bigger than Natural Number Set”; “the prodution line and formal language of looking for something belongs to an infinite set but is impossible to be found inside this infinite set (the confusing mechanism of ‘potential infinite--actual infinite’)” can not only be taken as “fatal fundamental defects” in present classical set theory resulting in the prodution of Russell’s Paradox, but also as “important fundamental theory” in present classical set theory resulting in the prodution of modern “Power Set Theorem” and “Uncountability of Real Number Set Theorem” , ...
Yous,
Geng Ouyang
Thank you for your suggstions dear Dr. Marcel Van de Vel, there are two different opinions between us. I discovered some fatal fundamental defects basing on my 40 years studies on “infinite paradox symptom complex” while you firmly believe that there is not “infinite paradox symptom complex” at all basing on the knowledge from books people (including you and me) have been educated:
1, In analysis: on the one hand, any one can use “the ‘potential infinite--actual infinite’ confusing formal language and pipeline operation” to construct all kinds of infinite related paradoxes; on the other hand, one can also use exactly the very same “formal language and production line” to construct all kinds of infinite related theorems. The typical example is: Zeno’s construction (proof) of “Achilles Can Never Chase Up Turtle Paradox” and its modern version of “creating infinite items each greater than 1/2 or 100 or 1000000000000000 or 1000000000000000000000000000000 or ... from the Un--->0 Harmonic Series and turn the Harmonic Series into a ‘Vn ---> any positive constants’ infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000)”-------one is called “paradox” but another is called “great mathematical theorem”.
2, In set theory: on the one hand, any one can use “the ‘potential infinite--actual infinite’ confusing formal language and pipeline operation” to construct all kinds of infinite related paradoxes; on the other hand, one can also use exactly the very same “formal language and production line” to construct all kinds of infinite related theorems. The typical example is: “the pipeline operation and formal language of looking for something belongs to an infinite set but is impossible to be found inside this infinite set (the confusing mechanism of ‘potential infinite--actual infinite’)” can not only be taken as “fatal fundamental defects” in present classical set theory resulting in the prodution of Russell’s Paradox but also as “important fundamental theory” in present classical set theory resulting in the prodution of modern “Power Set Theorem” and “Uncountability of Real Number Set Theorem” -------one is called “paradox” but others are called “great mathematical theorem”.
By the way, please don’t worry about so many simplified popular versions of Russell’s Paradox (such as "The Barber’s Paradox, The Liar’s Paradox, The Postcard Paradox,...) and, I would say "{0,2,4,...}＝{1,2,3,4,...}＞{1,3,5,..} paradox" is not an error in present classical infinite related mathematics at all-------it can be another important theorem as “Power Set Theorem” and “Uncountability of Real Number Set Theorem” in present classical infinite related mathematics.
Yous,
Geng Ouyang
Thank you for your ideas dear Dr. Marcel Van de Vel ·,
1,Do you mean Russel's paradox has nothing to do with mathematics and has nothing to do with “The Third Mathematical Crisis”?
2,Each time when I ask you the question of “how many items in Harmonic Series you operate scientifically for the first 100000000000000000000000000000, how many for the second 100000000000000000000000000000, how many for the third 100000000000000000000000000000, how many for the fourth 100000000000000000000000000000, how many for the fifth 100000000000000000000000000000, ... and turn the Un--->0 Harmonic Series into a Vn ---> any positive constants’ infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000)?”, I am very sorry to say that I never heard your answers------the truth is most people in the world since Zeno’s time (including you and 40 years ago of me) just firmly believe the idea from someone “we can turn the 'Un--->0' infinite Harmonic Series into a ‘Vn ---> any positive constants’ infinite series” but never know how to operate（The contemporary mathematics version of Hans Christian Andersen’s fairy tale " The Emperor's New Clothes " has been seeing frequently in present classical mathematical analysis？）.
3,Not only "{0,2,4,...}＝{1,2,3,4,...}＞{1,3,5,..} paradox", there are many more absurd examples telling us that it is a must for the “one-to-one corresponding operation" to have its basic theory and more scientific formal language (big, small, ... ).
4，If the fundamental defects of “potential infinite--actual infinite” are removed and the new basic theory of "abstract infinite law--the carriers of abstract infinite law" is developed for the infinite related mathematics, it is not only the matter of “alternative mathematics”.
Yous,
Geng Ouyang
Thank you for your ideas and patient dear Dr. Marcel Van de Vel,
For your Ad 1, Just one suggestion: please review how we accept the proof of Power Set Theorem with “So={x|x📷S x📷φ(x)}” in present classical set theory and, try to have an investigation on “The Third Mathematical Crisis”.
For your Ad 2, I am very sorry to say that I really never saw your answers to my question of “how many items in Harmonic Series you operate scientifically for the first 100000000000000000000000000000, how many for the second 100000000000000000000000000000, how many for the third 100000000000000000000000000000, how many for the fourth 100000000000000000000000000000, how many for the fifth 100000000000000000000000000000, ... and turn the Un--->0 Harmonic Series into a Vn ---> any positive constants’ infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000)?”. Would you please be so kind as to tell (or copy from somewhere you ever post) here what your answers are.
Yous,
Geng Ouyang
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)
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