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?
Mainly dealing with the “nature, existing condition, manifestation and relationship among each other” of elements in infinite sets as well as their inevitable relationship with the concepts of “potential infinite and actual infinite”, these two conundrums disclose an unavoidable fatal defect in present classical set theory-------the confusion of “impossible-scientific-defined potential infinite and actual infinite” have been making us human impossibly understand on earth whether the infinite sets we have been facing to are “potential infinite sets” or “actual infinite sets”, how to carry on quantitative cognizing activities to “potential infinite elements” or “actual infinite elements” , whether the elements’ numbers in infinite sets we have been facing to are “potential infinite many (much)” or “actual infinite many (much)”, ...? Following three are their specific performances:
（1）impossible to cognize scientifically the characteristics, the relationships of elements in different infinite sets (such as the elements in Real Number Set, Rational Number Set, Natural Number Set, ...) as well as the elements’ inevitable relationship with the concepts of “potential infinite and actual infinite” (if the elements are “potential infinite”, how to be “potential” and with what kind of “potential infinite elements’ characteristics”? if the elements are “actual infinite”, how to be “actual” and with what kind of “actual infinite elements’ characteristics”? are there different quantitative cognizing theories to “potential infinite elements” or “actual infinite elements”? ...).
（2）impossible to cognize scientifically the characteristics, the relationships of different infinite sets (such as Real Number Set, Rational Number Set, Natural Number Set, ...) as well as their inevitable relationship with the concepts of “potential infinite and actual infinite” (if the sets are “potential infinite”, how to be “potential” and with what kind of “potential infinite sets’ characteristics”? if the sets are “actual infinite”, how to be “actual” and with what kind of “actual infinite sets’ characteristics”? are there different quantitative cognizing theories to “potential infinite sets” or “actual infinite sets”? ...).
（3）impossible to cognize scientifically the relationships between and among “infinite sets” and “the elements in infinite sets”--------can we alter at will the “nature, existing condition, manifestation” of elements in infinite sets (such as alter the elements in infinite Even Number Set and infinite Odd Number Set and change both sets into Natural Number Set to prove they three have the same cardinality)? What kind of “potential infinite elements” or “actual infinite elements” will construct their relating “potential infinite sets” or “actual infinite sets”?
The typical paradoxes produced by the confusion of “potential infinite and actual infinite” in the foundation of present classical set theory by these two conundrums are different members of Russell's Paradox Family (such as The Story of Hilbert Hotel, the proofs of “Power Set Theorem” and “Uncountability of Real Number Set Theorem”).
Some people nowadays really believe that Russell's Theory of Types or ZFC really can solve Russell’s Paradox, but the facts proved this is impossible at all because neither Russell's Theory of Types nor ZFC can stop people applying exactly the same mechnisim of Russell’s Paradox (looking for some elements belonging to an infinite set but is impossible to be found inside this infinite set) to prove “Power Set Theorem” and “Uncountability of Real Number Set Theorem”.
In The Second Generation of Set Theory basing on the new infinite theory system, all the “infinite sets” and “elements in infinite sets” can only be “infinite related mathematical carriers” and all the quantitative cognizing theories and operations to them are only aiming at “mathematical carriers of abstract infinite law”-----nothing to do at all with “potential infinite and actual infinite”. The explicit quantitative cognized objects guarantee us never produce those quantitative cognizing conundrums and paradoxes generated and nourished by the confusion of “potential infinite and actual infinite” frequently appearing in the first generation of set theory. The same as in the mathematical analysis, only with the new theories of “infinite mathematical carriers and their relating qualitative--quantitative analysis and operation” can we really solve those suspended quantitative cognizing conundrums and paradoxes in “infinite paradox symptom complex” in The First Generation of Set Theory.
In my experience, actual infinity assumption is either a matter of faith or personal preference. I have only seen authoritative arguments to accept that infinity is actual. The only rational fact for its acceptance, is the successful existence of non-countable sets, based upon the Cantor's theorem, the proof of which is built upon the assumption of actual infinity.
Nevertheless, this is not an unavoidable situation. In this paper http://www.uav.ro/applications/se/journal/index.php/TAMCS/article/view/168,
you can see that the existence of on countable sets can be stated by local methods based on the information involved in definitions. In addition, by three paradoxes, It is shown that Cantor' s method sometimes proves nothing.
However, on the one hand, you can assume actual infinity as an axiom disregarding that it is an endless source of paradoxes. On the other hand, both Quantum Mechanics and Relativity Theory go in the opposite direction. Real world does not need such an axiom.
Our infinite related science history has proved that the concept of infinite is composed by both “potential infinite” and “actual infinite” since antiquity. On the one hand, no one is able to deny the qualitative differences and the important roles “potential infinite--actual infinite” play in the foundation of present classical infinite theory system; on the other hand, no one is able to deny that the present classical set theory is basing on “potential infinite and actual infinite” concepts as well as their related whole present classical infinite theory system. The fact is: any areas in present classical infinite related science system (of cause including present classical mathematical analysis and set theory) can not run away from the constraining of “potential infinite and actual infinite” concepts-------all the contents in present classical mathematical analysis and set theory can only be existing in the forms of “potential infinite mathematical things” and “actual infinite mathematical things”.
--------We human have been torture by “potential infinite” and “actual infinite” for thousands of years, we have to open up a new way to free us from this abysm.
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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