Cantor had told us clearly in his two famous diagonal disproofs that “the elements in real number set are more than those in natural number set” and “the elements in power set are more than those in its original set”: during the “one-to-one correspondence”, after the elements in natural number set and the original set have been finished up, the elements in real number set and power set are still a lot (infinite) remained------- the elements in real number set and power set are never-be-finished, endless, limitless and they are really infinite while those in natural number set and the original set are sure-be-finished, ended, limited and they are actually finite.

In fact, according to the ideas and operations in Cantor's Power Set Theorem, the quantity of infinite elements in all the infinite sets can be proved (turned into) finite, ended and limited. In present “potential infinite--actual infinite” based classical science system, Cantor's above two widely acknowledged proofs has proved another important theorem: many infinite sets in mathematics actually can be proved to be (turned into) finite set. Following the idea of “actual infinite” since Cantor's time, people have been looking for and proving the existence of “those infinite sets without containing infinite elements so they are smaller than the others”. If “Continuum Hypothesis” were proved to be right (scientific), more and more infinite sets would be turned into finite sets containing “sure-be-finished, ended, limited elements”. Following 8 unavoidable questions in “infinite small, infinite few, infinite great, infinite many” quantitative cognizing and conundrums in set theory have been challenging and bothering us human since Cantor's time:

1.       Does the definition of each “infinite set” closely relate to “the elements’ nature, appearance and inter-relationship” ---------the very characteristics of the elements inside the very “infinite set”?

2.       If it does not at all and the elements in all different infinite sets are the same a heap of “infinite abstract stuff without any differences and relationship”, then how can we define and distinguish “different infinite set” and how can we believe that there may be quantity differences between “different infinite sets”?

3.       If it does and the elements in all different infinite sets are “the concrete carriers of ‘abstract infinite concept’ with differences and relationship” (unique characteristics), then how will these unique characteristics decide the quantity differences between “different infinite sets”?

4.       Are “infinite sets” in present set theory “actual infinite sets” or “potential infinite sets”?

5.       Are infinite elements in infinite sets “actual infinite many” or “potential infinite many”? If they are “actual infinite many”, how can we conduct the quantitative cognitions to them; and if they are “potential infinite many”, how can we conduct the quantitative cognitions to them?

6.       What kind of mathematical tool of “one-to-one correspondence” is? When we conduct the quantitative cognitions to different infinite sets with “one-to-one correspondence” tool, is it “one element corresponding to one element” or “many elements corresponding to one element” or “many elements corresponding to many elements”, is it “potential infinite many elements corresponding to potential infinite many elements” or “actual infinite many elements corresponding to actual infinite many elements” or “potential infinite many elements corresponding to actual infinite many elements”?

7.       How can we define “infinite” and “finite” if we agree with the idea and the operations in Cantor's proofs that many infinite sets in mathematics actually can be proved (turned into) finite set-------the elements in real number set are never-be-finished, endless, limitless and they are really infinite while those in natural number set are sure-be-finished, ended, limited and they are actually finite?

8.         What kind of mathematical tool of “limit theory” is? When we conduct the quantitative cognitions to different infinite sets, how can we use limit theory to analyze, manifest and treat those X--->0 elements’ number forms inside them (such as those X--->0 elements’ number forms in [0, 1] real number set)?

The formulation of this question is puzzling.

I write puzzling, since it not clear what you mean by your assertion that the set of natural numbers is sure-[to]be-finished, ended, limited.   Perhaps this will become clearer, if you explain each of these terms applied to the set of natural numbers.

 The question contradicts the fundamental principle of thought process and one of the basic axioms of ontology, something either "is" or " it is not", but not both. Based on the definition of a set being finite or infinite, if a set is finite then it can not be infinite and if it is infinite then it can not be finite. No finite set that is equivalent to any of its proper subsets and hence can not be infinite. Besides, the set of natural numbers do not end or finished as you have said (if that is what you meant) 

Dear Dr. James F Peters and Dr. Dejenie A. Lakew, thank you very much for your ideas.

My question came from another question: How can we know real number set has more elements than natural number set and why?

Would anyone please give his/her answer to the question of “How can we know real number set has more elements than natural number set and why”------- if ignoring the fact that the elements in real number set are more than those in natural number set because during the “one-to-one correspondence” quantitative cognizing operations, after the elements in natural number set have been finished up, the elements in real number set are still a lot (infinite) remained?

Thank you again.

Yours sincerely, 

Geng

It is true that many people say we human in fact can have as many definitions to “infinite” as we like in our science so can solve above “infinite-finite” conundrum easyly. Is it possible and scientific?

It may have something to do with “Philosophy and Mathematics”? I think for Philosophy work, it is more abstract than Mathematics; while for Mathematics work, it is more concreteness than Philosophy. For example: for Philosophy, it is enough to say “A is heavier than B” but for Mathematics we need to say “A is 9576 grams heavier than B”.

It is also typical in Harmonic Series Paradox: for Philosophy, it is enough to say “we are able to produce infinite items each bigger than 1/2, or 100, or 1000000, or 1000000000000000000,…by brackets-placing rule with limit theory from Un--->0 items in Harmonic Series and change Harmonic Series into an infinite series with infinite items each bigger than any positive constants (such as 100000000000000000000000000000)”; but for Mathematics we need to know “how many Un--->0 items in Harmonic Series we use to produce the first positive constants (such as 100000000000000000000000000000) and how many for the second…

In the axiomatic approach  of set theory every set has a cardinal number.Two sets have the same cardinal number if there is a one to one correspondence between them. For any natural number n,the set {1,2, ...,n} has the cardinal number n.A set is finite if it is empty or has a cardinal number n,otherwise it is infinite.Cardinal numbers of infiinite sets are not natural numbers.The set of cardinal numbers has a natural ordering and is infinite according to the well known Cantor theorem,namely that,for any set A , the cardinal number of the power set of A is greater than the cardinal number of A.The first infinite cardinal number is that of the set of natural numbers.It is known that the cardinal number of the set of real numbers is that of the power set of natural numbers,whence it is greater than the cardinal number of this set.

Dear Dr. M.El-Ghali Mohamed Abdallah, thank you very much for your ideas.

The fact is that during the “one-to-one correspondence” operation between Real Number Set and Natural Number Set, after the elements in Natural Number Set have been finished up, the elements in Real Number Set are still a lot (infinite) remained:

 1, The elements in real number set are never-to-be-finished, endless, limitless------Real Number Set is really infinite!

2, The elements in natural number set are sure-to-be-finished, ended, limited-------- Natural Number Set is actually finite?!

A typical case is Cantor's Power Set Theorem: any infinite set can be turned into finite set in front of its own Power Set-------because during the “one-to-one correspondence” operation between the original set and its power set, after the elements in the original set have been finished up （finite）, the elements in its own Power Set are still a lot (infinite) remained!

-------If the elements in a set are sure-to-be-finished, ended, limited, does it called finite set or infinite set?

Thank you again.

Yours sincerely, 

Geng

Nine of my published papers have been up loaded onto RG to improve the work:

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) --------The Mistakes and Defects in the Classical Definition of “Infinite” and the 2500--year Obstacles in Quantitative Cognitions to “Infinite Things”

https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II

3，On the Quantitative Cognitions to “Infinite Things” (III) ---------Fourteen Challenging Conundrums of Infinite Things’ Quantitative Cognitions Produced by the Absence of the “Infinite Carriers’ Theory” and the Confusion of “Potential Infinite” and “Actual Infinite”

https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III

4，On the Quantitative Cognitions to “Infinite Things” (IV) ---------“Infinite”, “the Infinite Mathematical Carriers” and “the Infinite Number Forms” in Mathematical Analysis

https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV

5， On the Quantitative Cognitions to “Infinite Things” (V) ---------Solving eight challenging “quantitative cognition” conundrums in present infinite set theory

https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V

6， On the Quantitative Cognitions to “Infinite Things” (VI) ---------A New Way Solving Fatal Defects in the Foundations of Present Classical Set Theory and Its Quantitative Cognizing Tool of “One to One Correspondence”

https://www.researchgate.net/publication/331428737_On_the_Quantitative_Cognitions_VI

7， On the Quantitative Cognitions to “Infinite Things” (VII) ---------The Underlying “Self-refutation Mechanism” Origin of “Infinite Paradoxes Syndrome” and an Exactly Same Paradoxical Phenomenon in Present Set Theory and Mathematical Analysis

https://www.researchgate.net/publication/335110516_On_the_Quantitative_Cognitions_to_Infinite_Things_VII

8， On the Quantitative Cognitions to “Infinite Things” (VIII) ------------ Three Exactly Same Fundamental Characteristics in the Old and New Generations of Infinite Set Theory and Mathematical Analysis

https://www.researchgate.net/publication/341754494_On_the_Quantitative_Cognitions_to_Infinite_Things_VIII

9， On the Quantitative Cognitions to “Infinite Things” (IX) --------- "The Infinite Carrier Gene”, "The Infinite Carrier Measure" And "Quantum Mathematics”

https://www.researchgate.net/publication/344722827_On_the_Quantitative_Cognitions_to_Infinite_Things_IX_---------_The_Infinite_Carrier_Gene_The_Infinite_Carrier_Measure_And_Quantum_Mathematics