Working in classical infinite theory system with the fundamental defect of “’potential infinite--actual infinite’ confusion”, any things in mathematics relating to “infinitesimals, infinities, infinite manies” can be “potential infinite” or ”actual infinite” even both “potential infinite and actual infinite”, we can see some typical examples from Cantor’s work. For instance, if you prefer to work in a “potential infinite way”, you just assert emphatically that the elements in all infinite sets are piles of infinite “abstract mathematical things (such as geometry points) with same existing law, same existing meaning, same existing forms, same relationship as well as same quantitative meanings, and you can prove that Infinite Rational Number Set is equal to Infinite Natural Number Set (the ideas and operations of proving all the proper subsets equal to their mother set are typical examples of “potential infinite way”), because Cantor believed that elements in many infinite sets are all with same existing law, same existing meaning, same existing forms, same relationship as well as same quantitative meanings; but if you prefer to work in an “actual infinite way”, you just assert emphatically that the elements in infinite sets are all with very clear differences of existing law, very clear differences of existing meaning, very clear differences of existing form, very clear differences of their relationship as well as very clear differences of quantitative meaning, and you can prove that Infinite Rational Number Set is bigger than Infinite Natural Number Set: The elements of tiny portion of rational numbers from Infinite Rational Number Set (such as the sub set : 1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) map and use up (bijective) all the numbers in Infinite Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so，infinite rational numbers (at least 2,3,4,5,6,…n,…) from Infinite Rational Number Set are left in the “one—to—one element mapping between Infinite Rational Number Set and Infinite Natural Number Set (not the integer set) ------- Infinite Rational Number Set has infinite more elements than Infinite Natural Number Set and they are not bijective at all (the ideas and operations of Cantor’s “Transfinite Numbers theory”, “continuum hypothesis”, “uncountable Real Number Set proof”, as well as “Power Set Theorem proof” are typical examples of “actual infinite way”) , because Cantor suddenly changed his idea and turned to believe that the elements in many infinite sets are all with very clear differences of existing law, very clear differences of existing meaning, very clear differences of existing form, very clear differences of their relationship as well as very clear differences of quantitative meaning.
Sorry, Geng Ouyang , I tried, but I just don't follow you.
Sets cardinalities, Cantor's stuff, etc, all that has been thoroughly cleared and worked out, and it looks quite simple and straightforward to me. There are no "potential" and "actual" infinities; actually the colloquial term "infinity" has little use in professional math. Also, there are many known examples of bijections between natural numbers, rational numbers and integer numbers, in any combination, and there are no mysteries accompanying those. The three sets are just equi-cardinal, end of discussion. Btw, I also object to the statement that "Cantor suddenly changed his idea and turned to believe ...". He was a mathematician and mathematician do not change ideas, nor do they believe anything; they may just become personally frustrated when they can not prove nor disprove a particular conjecture for a long time (it keeps the conjecture at a level of folklore and completely outside of mathematics).
Thank you dear Mr. Stan Sykora，
1) I am sorry that I am unable to find a suitable statement for my idea that the fundamental defect of “’potential infinite--actual infinite’ confusion” deeply infects people (including Cantor) who have been working in those fields relating to infinite.
2) it is true that the concepts of "potential infinite" and "actual infinite" have been in our science along with the concept of "infinite" coming to being. It is not that easy to get rid of them from our science.
Dear Geng. I see that our dialogues have given you new perspectives as well as me, and that it shows that you have started a new topic of debate.
From the dialog "Infinite Rational Number Set has more elements than Infinite Natural Number Set! Is anything wrong?"
Thank you dear Mr. Carlos Oscar Rodríguez Leal，
The idea of this topic also comes from my papers On the Quantitative Cognitions to “Infinite Things” (IV) and On the Quantitative Cognitions to “Infinite Things” (VI) being uploaded on RG.
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