There have been three suspended questions in present classical set theory ever since:
1, the definition of set------should the definition of set concern the nature of elements inside the set? How we distinguish different sets（Such as Odd Number Set and Natural Number Set）? Will the nature of elements inside the set decide the existing state of the set as well as its relationship with other set?
But, it is “the different natures of the elements in real number set and natural number set” that make Cantor proved the different cardinalities between the two sets.
2, how to judge whether a set belongs to “potential infinite set” or “actual infinite set” or both “potential infinite set” and “actual infinite set”? What kind of nature do the elements have inside “potential infinite set” or “actual infinite set” or both “potential infinite set” and “actual infinite set”?
3, can we have many different bijection proofs with different result between two infinite sets? If we can, what conclusion should people choose in front of two opposite results, why?
Some people think the nature of the elements in a set are in no way related to the cardinality.
But in fact, it is “the different natures of the elements in real number set and natural number set” that make Cantor proved the different cardinalities of the two sets.
Dear Geng,
1. A set is a well defined collection of objects. So, it should define the nature of the set. Always you can find a relation between any two sets. For example, intersection is a binary relation between any two sets, which could output either an empty or non-empty set. In case of Even and Odd their intersection is empty.
2. As far as I understand, infinite means unbounded or has no ending regardless of set or series or sequence. It is clear and well defined. I am not able to find proper well defined definitions for potential infinite or actual infinite. Also, I am not able to distinguish them. I am sorry to say this, as they are not defined well, the debate on potential and actual infinite will lead to an infinite debate, for which again we have to debate whether that infinite debate is potential or actual.
3. If you consider, any two finite sets of same cardinality n, then there are n! (n factorial) bijections possible. Is it not allowed to have more bijections between any two finite sets? Is it an offensive thing? I hope you won't say "it is not allowed in finite case". When it is possible to find n! bijections for finite sets, why don't we consider different bijection proof for infinite sets. If there exists a bijection between two infinite sets, definitely, there is an infinite bijections. So, you can choose any bijection between two sets. (Note that between two sets, not subset of domain and codomain)
Dear Dr. Panchatcharam Mariappan, thank you.
1, can we really define a set (a well defined collection of objects) without the nature of elements inside the set?
2, when working in present “potential infinitesimal--actual infinitesimal” based infinite related science system, I felt uneasy and even horrible sometimes-------every “fruit” on a confused foundation of “potential infinitesimal--actual infinitesimal”?! It was the very reason that powered me to start a long and hard journey 40 years ago on the “new foundation of infinite related science system”. I am sure this work should be done sooner or later if human still alive and need “science”.
3, I agree with you that there can be many different bijection proofs between two infinite sets. But the things worry me is “what conclusion should people choose in front of two opposite results, why?”
Best regards,
Geng
Dear Geng
1. Well defined implies that, we can understand the nature of elements. Mathematically, it is not necessary to explain each element as a an explicit function of a variable. For example, prime number. Also, it is not necessary and sometimes not possible to write a complete list of elements. For example, any infinite sets.
2. From my knowledge, the debate is there for a long time. But, most of the set theoretical, calculus and applied mathematics are related the theory well developed by Cantor and their successors. So, far we don't have any contradictory observations based on this development. If one can find a flaw, definitely a basic system would require a change. Also, we should agree that not everything in the universe could be explained by our current science.
3. Unless there are no contradictory results one can choose any bijection proof. What do you mean by "opposite results"?
Dear Dr. Panchatcharam Mariappan, thank you.
1, what are the natures of elements in sets? How can we understand the nature of elements in a set? How we distinguish different sets（Such as Odd Number Set and Natural Number Set）?
2, if “the debate is there for a long time and will be endless”, it discloses something wrong somewhere in human science. It is a challenge for researchers?!
3, by "opposite results", I mean followings:
(1) the way of my bijection proof: 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 has infinite more elements than Natural Number Set and they are not bijective at all.
(2) Cantor’s diagonal way to prove that Rational Number Set has same elements as Natural Number Set.
Sorry，I will be back in 8 days after a business trip.
Best regards,
Geng
1. You can understand the nature of elements of a set by its definition only. For example. collection of boys in a class is a set. So, here the nature of the set is that its elements are all boys. To distinguish sets, we have to understand the nature of elements. In even number set all elements are even, that means divisible by 2, whereas in odd number set elements are not divisible by 2.
2. I am not sure about it, but if you consider both infinity as the meaning of unbounded, none of our existing theory would fail. If any of our theory would fail due to this fact, human will think of new base or new definition. Necessity is the mother of invention. I think there is no necessity so far. (may be I am wrong)
3. The nature of infinite set is that it can have bijective map with its own subset. So, there is no harm in it. Also, they are not contradicting anything, as you have not provided any bijective map between N and Q, that map is not a bijective map between N and Q, instead it is a map between N and a subset of Q. Also, I would like to add one more point, if there are more than one bijection, they wont produce any opposite results at all. By seeing this point, I remember a joke. A school kid come and say to her mom that maths teacher is wrong, because yesterday she said 4+5=9, and then 6 + 3 = 9. Today she says that 7+2=9. If you consider "opposite results" with this you will definitely feel the argument is not correct.
Hello everybody.
- The definition of set can be considered to be a still open issue. People adopted ZFC as a good compromise, but a lot of people are not happy with this. First at all, we don't know if ZFC is consistent or not, and probably we will never do. Then a lot of people do not consider sentences as ""CH is independent of ZFC"" as answers for the problem. They think that a question like the Continuum's Hypothesis must have a clear answer.
- The situation is completely different with actual vs. potential infinity. Those attributes "actual and potential" are not about the set, and even less about its infinity. It is described only how the fact that a set is infinite does interfere with the discourse about the set, with the demonstration using this or more generally, with the context of this statement. They are contextual qualities and not intrinsic qualities. The same infinite set can be used as a potential infinity in some proof and as actual infinity in another proof.
- Moreover, a set is not potentially infinite or actually infinite. The set is finite or infinite. Even less probably you can define a set which is potentially infinite but not actually infinite. Again, those attributes refer to the context and not at all to the set.
Thank you, everybody!
So, we have another question: how many infinite things can run away luckily from "potential infinite and actual infinite" in mathematics and science as well (narrowly and broadly )?
Dear Dr. Mihai Prunescu, Dr. Miguel Ángel Montes and Dr. Waldemar Koczkodaj, thank you.
"Potential infinite and actual infinite" have been used for "infinite" ever since. But, the very trouble is: people accept “potential infinity” and “actual infinity”, but no one so far in the world can provide the proper definition for potential infinity and actual infinity and people have been keeping arguing and debating on the natures of them ever since-------endless and unsolvable.
This is one of the fatal defects in present infinite related classical mathematics system： no one knows what is in front of you when treating “infinite things”--------“potential infinite” or “actual infinite”, how and what to do with different infinity?
Can any infinite things in present classical "infinite" related science system be excluded from "potential infinite and actual infinite"?
Best regards,
Geng
To answer your question related to the trouble (which you consider), let me ask you a question. Please answer this question frankly and honestly. How many degrees is the earth tilted? How?
Dear Dr. Panchatcharam Mariappan, I am very sorry to say that I really don’t care “How many degrees is the earth tilted?”. So, I am unable to answer your question.
Best regards,
Geng
Dear Geng,
No problem. My answer is somehow related to the answer to the question.
You have mentioned that "
"no one so far in the world can provide the proper definition for potential infinity and actual infinity and people have been keeping arguing and debating on the natures of them ever since-------endless and unsolvable."
Without proper definition, how is it possible to give a concrete answer. Because of this reason, it is better to fix with a single definition of infinity by omitting potential or actual, where infinity means unbounded. In other words, sets which are not finite is infinite.
If we want to change the basic definition of a concept, it could affect the entire system and sometimes it may collapse. For example, we use the fact that the earth is tilted by 23.5 degree (approximately). It is based on an assumption that "When the fingers of the right hand are curled around in the direction of the planet's rotation, the thumb points in the direction of the north pole". If we change the point of direction to south pole, definitely the tilting angle will be different. Also, if we consider the world map, mostly north pole on the top and south pole on bottom. Why don't we keep the map upside down and discuss?
Of course, no one stop you to change the point of direction or map. We can do a different study on this and develop a theory, but that should be well defined.
In the same fashion, you are welcome in the research world with innovative ideas to define "potential inifinity and actual infinity or Geng's infinity or Zeno's infinity or anynamed infinity as you wish". But that should be well defined. Until it is well defined the research/science community won't accept that. At the same time it won't deny or degrade the researcher, but waits for the fruitful outcomes. If you could redefine or find a proper definition for actual infinity and potential infinity and distinguish them properly, develop a theory based on it, science community will definitely accept it. All the best for your research.
Dear Dr. Panchatcharam Mariappan, thank you.
I totally agree with you that "it is better to fix with a single definition of infinity by omitting potential or actual, where infinity means unbounded. In other words, sets which are not finite are infinite." This is the very idea I am now working with ------ theoretical infinite and practical infinite.
In new infinite system, there is only one definition of infinite: unbounded (somewhat like that of potential infinite in present traditional science system) and a systematic new “infinite carrier” theory (somewhat like that of actual infinite in present traditional science system) will be pioneered and developed. So, in most cases, we are facing and working with “infinite carriers” but not “potential or actual infinite things” in analysis and set theory.
For at lest 2500 years since Zeno's time, in the “cognizing area of infinite”, people have been trapped deeply in the confusing vortex of “abstract concept and its carriers (such as “fruit” and “apple, banana,…”)” and this ill situation should be changed by some people, not matter who, sooner or later------this is the natural metabolism law.
Thank you very much again Dr. Panchatcharam Mariappan.
Best regards,
Geng
It is true that people have been acknowledging the concepts of “potential infinite” and “actual infinite” are the foundation of present classical infinite related science and mathematics theory systems since antiquity. This situation decides all of our “Infinite things cognizing work” are affected and garmented unavoidable by the concepts of “potential infinite--actual infinite” and there must be “potential infinitesimals--actual infinitesimals”, “potential infinities--actual infinities”, “potential infinite sets--actual infinite sets”, “potential infinite things--actual infinite things”, … in human science and mathematics theory systems. But, because of not knowing clearly the “ontology--form” natures as well as the differences of “potential infinite--actual infinite” and unable to understand what kind of infinite we are facing to and treating with, many successful fruits have been achieved accompanying with many paradoxes and mistakes in our “infinite things cognizing work.
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