The fundamental defects of “potential infinite and actual infinite” confusions in present classical infinite set theory have been making us humans unable to study and cognize scientifically the foundation of “one-to-one correspondence theory” (the “one-to-one correspondence theory needs its own foundation” even have never been considered about). And, because of the absence of this very foundation, it is very difficult for people to really understand scientifically what kind of mathematical tool “one-to-one correspondence theory” is and how to operate with this mathematical tool in practical quantitative cognitions to elements in infinite sets. So, following five questions have been produced and troubling people long:
（1）Are the elements in infinite sets “potential infinite things” or “actual infinite things”？
（2）Are there different “one-to-one correspondence theories and operations” to “potential infinite elements” or “actual infinite elements” in infinite set theory？
（3）How do we practically carry on “one-to-one correspondence” operations between two sets-------do we have “‘one single element’ to ‘one single element’ correspondence” or “‘one single element’ to ‘many elements’ correspondence” or “‘many elements’ to ‘many elements’ correspondence”？can we arbitrarily alter the elements’ “special nature, special existing condition, special manifestation and special relationship among each other” in infinite sets during the “one-to-one correspondence operations” for quantitative cognitions (such as alter all the elements in Natural Number Set first [1x2, 2x2, 3x2, 4x2, …,nx2, …] = [2,4,6,8, …,e,…] (not the correspondence between N and E but E andＥ), then prove it has same quantity of elements in Even Number Set)?
（4）What kinds of the elements in two different infinite sets are corresponded-------- do we have “‘one single original element’ to ‘one single original element’ correspondence” or “‘actual infinite elements’ to ‘potential infinite elements’ correspondence” or “mixture correspondence of ‘actual infinite elements’ and ‘potential infinite elements’”？
（5） What on earth is the foundation of “one-to-one correspondence theory”？
The fundamental defects in present classical infinite set theory have made us unable at all to answer clearly and scientifically above five questions. So, when carrying on practical quantitative cognitions to elements in different infinite sets with “one-to-one correspondence theory”, one can do very freely and arbitrarily--------lacking of scientific basis. For example: it is because of acknowledging the differences of elements’ “special nature, special existing condition, special manifestation and special relationship among each other” between Real Number Set (R) and Natural Number Set (N), one can prove that the Real Number Set (R) has more elements than N (the Power Set Theorem is proved in the same way). But, as what has been discussed in above 2.1 .1, we are able to prove with exactly the same way “the mother set has more elements than its sub-set”, “Rational Number Set has more elements than Natural Number Set”, “Natural Number Set has more elements than odd number set” ,...; we can even apply the widely acknowledged method of altering elements’ “special nature, special existing condition, special manifestation and special relationship among each other” to prove “Natural Number Set has more elements than Natural Number Set”, “odd number set has more elements than even number set”, “even number set has more elements than odd number set”, ....
Basing on the new infinite theory system with the “infinite mathematical carriers theory”, the Second Generation of Set Theory provides us with the scientific foundation of “one-to-one correspondence theory” and enable us answer above five questions clearly and scientifically:
（1）the elements in infinite sets are “infinite related mathematical carriers” with explicit quantitative nature and definition, indicating the existing of “abstract infinite law” and nothing to do at all with “potential infinite--actual infinite”. This decides one of the major differences between the first and the second generation of set theories-------the elements in different infinite sets have their own “special nature, special existing condition, special manifestation and special relationship among each other. So, it is really possible that different infinite sets have different quantity of elements and people can take them really as “visible and tangible infinite related mathematical things (such as the new numbers in new number spectrum)” for the quantitative cognitions
（2）the elements in infinite sets have nothing to do at all with “potential infinite elements” and “actual infinite elements”, there is only one identity for them-------“infinite related mathematical carriers” with explicit quantitative nature and definition; So, there is only one “one-to-one correspondence theory and operation” for them.
（3）it is explicitly stipulated that only “‘one single original element’ to ‘one single original element’ correspondence” operation is scientific (allowed) when comparing two sets for the quantitative cognitions and, during this process, any operations of arbitrarily altering the elements’ “special nature, special existing condition, special manifestation and special relationship among each other” are unscientific (not allowed).
（4）in the Second Generation of Set Theory, because of nothing to do at all with “potential infinite--actual infinite”, it is impossible to have any troubles produced by the confusion of “potential infinite --actual infinite”.
（5）the new infinite theory system (especially its theory of “infinite related mathematical carriers”) is the foundation of “one-to-one correspondence theory”.
Thank you dear Dr. Peter Breuer
1, “It is just squiggles on paper, nothing more.”
But, with those “squiggles on paper”, power set theorem is right there which generates many of the problems.
2, So many people have been trying to run away from “potential infinite--actual infinite” ever since “infinite” came into human science (mathematics). It is impossible because these two concepts are deeply rooted in “infinite” --------infinite law and the carriers of infinite law.
3, Without a sound foundation of scientific infinite theory system, Non-Standard Analysis, Type Theory and Model Theory (including ZFC) can do nothing to help humans solve those suspended infinite related paradoxes produced by the confusion of “potential infinite-actual infinite” in present mathematical analysis and set theory.
I wish you are able to understand what I said above and I beg your pardon for my very poor English as English in my second language.
Yours,
Geng Ouyang
Thank you dear Dr. D. Singh,
But, basing on classical infinite theory system with “potential infinite--actual infinite”, the Intermediate Set Theory can do nothing to help humans solve those suspended infinite related problems and paradoxes produced by the confusion of “potential infinite-actual infinite” in present set theory.
Yours,
Geng Ouyang
Thank you dear Dr. Peter Breuer
1, I am talking about “Is the proof of Power Set Theorem scientific or not”, it is nothing to do with “whether I like (if I like to use) Power Set Theorem or not”.
2, l think Logic-Structure-Intuition can not be taken apart, they are all necessary in humans’ scientific work. I apply them happily in my research work.
3, I don’t know how long the time might leave for me in this world and I really have not the desire to “Become a Gengist”. I just try my best to disclose the unscientific things in “infinite related science and mathematics” although I knew very clearly from very beginning 40 years ago that this will lead me to a lonely life because I was against the classical infinite system with more than 2500 years history.
I wish again you are able to understand what I said above and I beg your pardon for my very bad English as English in my second language.
Yours,
Geng Ouyang
Dear Geng,
Yes, I agree that the concepts of (infinite) or infinity, infinite sets, everything related to infinity !! is confusing at first glance.
In the first course of calculus, a student faces
0 x infinity does not equal 0,
infinity - infinity is not 0
the infinite sum of zeros 0+0+0+0+........ is not ,0 !!!!!
Ah, infinity is not real and behaved differently with the usual operations.
BUT,
The in-depth study of such phenomena creates different branches of mathematics: real and mathematical analysis ( sequences of numbers and convergence, etc..), Topology (neighborhoods, homeomorphisms, Hausdorff spaces, compact spaces, etc..). It is a long, long story of the development of human knowledge to describe our surrounding universe, starting from Euclidean postulates to the relativity of Einstein.
I raise the following question: Can the part include the whole?
The Surprising Answer: Sometimes Yes.
In fractals: If we take a tinny image (epsilon - neighborhood) of the whole picture, we obtain the image of the whole picture itself !!!
In set theory:
We said that the set A is equivalent to the set B if and only if, there exists a bijection from A to B.
And if A is equivalent to any of its proper subsets, it is called INFINITE SET.
In your example f (x) = 2x,
f is a bijection from N into E ( natural numbers to even numbers),
but E is a proper subset of N, we said that N is infinite. Why not?
It makes sense and compatible with the definition.
In topology, R is homeomorphic to an interval of real numbers (a, b )
i.e., there exists a continuous bijection from R into (a,b), and (a,b) is a proper subset of R.
No contradictions at all everything goes smoothly with logical flow into all branches of mathematics.
Thousands of mathematicians through the history of mathematics tried to show the fifth axiom of Euclid's ( Parallel postulate: parallel lines in the plane not intersecting as prolongated to infinity).
No one succeeded and the result was the creation of several non-Euclidean geometries. Klein invents his famous mathematical model to show that all such geometries ( Elliptical, Hyperbolic, etc. ) are consistent and no contradictions at all.
I am optimistic; real numbers are nice, everything is logical, and our world is consistent.
Best regards
Thank you very much dear Dr. Issam Kaddoura,
Following infinite related questions have never been answered scientifically since the concept of “infinite” came into human science:
Are the concepts of “potential infinite--actual infinite” important in present classical infinite related science system? If yes, what roles they play; if not, why they have been existing in our science ever since? How can we cognize the relationship between “infinite related mathematical things” and “potential infinite--actual infinite”?
So, following thousands-year old contradictions have been inevitably existing in present infinite related set theory and mathematical analysis:
（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 (the Second Mathematical Crises triggered by the Berkeley Paradox has never been solved at all; on the other hand, one can also use exactly the very same “formal language and production line” to construct all kinds of infinite related “important mathematical proofs and 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: on the one hand, any one can use “the formal language and production line operation of looking for some elements belonging to an infinite set but impossible to be found inside this infinite set” caused by the confusion of “potential infinite--actual infinite” 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 operation” to construct all kinds of infinite related “important mathematical proofs and theorems”. The typical example is: the construction of “Russell’s Paradox” and its modern versions of fundamental “Power Set Theorem proof” and “Uncountability proof of Real Number Set”.
Yours,
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)
Dear Geng Ouyang ,
Well done Geng.
Could you please download the links.
Best regards
Thank you dear Dr. Issam Kaddoura,
From the following links https://www.researchgate.net/profile/Geng_Ouyang2/publications
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
Best regards
Thank you dear Dr. Peter Breuer,
I just sincerely tell you that my 45 years work in above five papers are far more than “part of an infinite series”.
Best regards
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