We understand from our infinite related science history that in present classical infinite related science system, it has been admitted that the concept of infinite is composed by both “potential infinite” and “actual infinite”. 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--actual infinite” concepts as well as its 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--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”. But, the studies of our infinite related science history have proved that no clear definitions for these two concepts of “potential infinite--actual infinite” and their relating “potential infinite mathematical things--actual infinite mathematical things” have ever been given since antiquity, thus naturally lead to following two unavoidable fatal defects in present classical set theory:
（1）theoretically: It is impossible to understand what the important basic concepts of “potential infinite” and “actual infinite” and their relating “potential infinite number forms, potential infinite sets” and “actual infinite number forms, actual infinite sets” are and what kinds of relationship among them are. So, in many “qualitative cognizing activities on infinite relating mathematical things (such as all kinds of infinite sets, elements in infinite sets, numbers of elements in infinite sets)” in present classical set theory, many people even don’t know or actually deny the being of “potential infinite” and “actual infinite” concepts as well as their relating “potential infinite number form, potential infinite sets” and “actual infinite number forms, actual infinite sets”--------it is impossible at all to understand clearly and scientifically the exact relationship among the important basic concepts of “infinite, infinities, infinite many, infinitesimals, infinite sets, elements in infinite sets, numbers of elements in infinite sets”, ... So, it is impossible at all to understand clearly and scientifically all kinds of different infinite sets (such as lacking of the “’set spectrum’ for the overall qualitative cognitions on the existing forms of infinite sets”), elements in an infinite set (such as ”are the infinite related elements potential infinite mathematical things or actual infinite mathematical things, how they exist?”), numbers of elements in an infinite set (such as ”are they actual infinite many or potential infinite many?”), the “one-to-one corresponding theory and operation” in infinite sets (such as ”are the potential infinite elements corresponding to potential infinite elements or actual infinite elements corresponding to actual infinite elements or actual infinite elements corresponding to potential infinite elements?”) ,... --------the unavoidable defects of qualitative cognition on infinite sets and their elements
（2）operationally: First, it is impossible to understand whether the “elements in an infinite set, numbers of elements in an infinite set and all kinds of infinite sets” being cognized in present classical set theory are “potential infinite mathematical things” or “actual infinite mathematical things”, whether there are different theories and operations for “potential infinite mathematical things or actual infinite mathematical things”, and it is impossible at all to understand correctly (scientifically) in present classical set theory the natures of infinite related quantitative cognizing theories and tools (such as limit theory and the “one-to-one corresponding theory”) and their operational scientificities-------- it is impossible at all to master correctly (scientifically) the operational competences and skills of limit theory and the “one-to-one corresponding theory” thus resulting in no scientific guarantee for the operations of limit theory and the “one-to-one corresponding theory”; second, it is impossible at all to judge the scientificities of many infinite related quantitative cognizing activities in present classical set theory, people in many cases can only parrot every bit of what have been done by others or do as one wishes to treat many “not—knowing—what” infinite mathematical things with the unified way of “flow line” (any “infinite sets”, “elements of an infinite set”, “elements’ number of an infinite set” can either be “potential infinite” or “actual infinite”, neither be “potential infinite” nor “actual infinite”, first “potential infinite” then “actual infinite”, first “actual infinite” then “potential infinite”, ,,,), those believed and accepted Russell’s Paradox, Hilbert Hotel Paradox, Cantor’s operations of “cutting an infinite thing into pieces to make super infinite numbers with different grades” as well as the famous “using Russell's ‘paradox constructing mechanism’ to prove the Power Set Theorem and the uncountability of real number set” are typical examples of “potential infinite--actual infinite” confusing operations--------the unavoidable defects of quantitative cognition on infinite sets and their elements.
For me it is relatively clear. Talking about big integers, say, potentially infinite means as large as necesary to use in a particular problem, and which can always be made larger if needed.
The actual infinite means just that.
regards, juan
In my view:
1" Potential infinite" means that we keep adding smaller and smaller quantities which become zero. In problems of this case we usually start with finite values on RHS ,, famous Zeno problem the total distance is fixed and then we divide it into smaller additions so we could be eventually adding only zeros.
2."Actual infinity" is when the additions do not really become zero. In such cases we really do not know the RHS and finding limits in such cases is a question of whether these additional quantities eventually become zero .
3 Consider also definition of point as space with zero dimension . It does not really make sense to add many many such points to construct a line segment.A point with finite dimension > zero however small can be used to construct a line segment. This corresponds to impossibility of finding a physical object with zero dimension.
4. "As far as the laws of mathematics refer to reality they are not certain and as far as they are certain they do not refer to reality" Albert Einstein (address to Prussian Academy of Sciences 1921)
Cheers
yes very interesting... At first glance to understand infinity actual or potential are not easy. For example on the real line betweeen two so so close numbers there are infinitely many numbers....
Thank you my dear colleagues.
This problem has been troubling us human since Zeno’s time: when facing “X--->0 number forms” in our quantitative cognizing operations in infinite related mathematical case, no one knows when they should be potential infinitesimals and when they should be actual infinitesimals and parroting is the only way. There is no systematic quantitative analysis and processing theory at all for the “ X--->0 potential infinitesimal or actual infinitesimal number forms”! So, for any infinite related numbers with the form of X--->0 (whether "potential infinitesimal or actual infinitesimal" or whether "could make it zero, take its limit, take the standard number" ), shall be treated in accordance with the specifications laid down by some traditional pipelines (this is the very reason why three generations of mathematical analysis completely equivalent to each other ).
The newly discovered Harmonic Series Paradox is a typical example: the confusion of “potential infinite, actual infinite” contents in present classical mathematical analysis resulted by the confusion of “potential infinite, actual infinite” concepts in present classical infinite theory system have been making people unable (impossible) to know what those “X--->0 number forms (items)” are in newly discovered Harmonic Series Paradox and even more, impossible to know whether they are potential infinitesimals or actual infinitesimals and how to treat them. Studies proved that the so called “infinitesimal” in present classical mathematical analysis are in fact a mixture of two “infinite mathematical carries (number forms)”: “X--->0 intersmall” and “X--->0 infinitesimal” . In many practice, one cannot distinguish at all the two different kinds of "mathematical carrier" and can only abuse of the limit theory operationally and theatrically-------it is wrongful (unscientific) to treat abusely "two number forms of Un--->0 with different natures" in Harmonic Series in a pipeline operation of "bracketing” by limit theory to create infinite numbers greater than 1/2 or 100 or 1000000000000000 or 1000000000000000000000000000000 or ... and turn the Un--->0 Harmonic Series into a “Vn ---> any positive constants” infinite series (with infinite items each bigger than any positive constants, such as 100000000000000000000000000000). The newly discovered Harmonic Series Paradox has become a typical contemporary mathematical version of Andersen’s fairy tale "the emperor's new clothes" and, the suspended Ancient Zeno’s challenging statement of “not matter how fast Achilles runs, he will never catch up with the slowly crawling Turtle in Achilles--Turtle Race” has been confirmed with exactly the same operations and same ideas basing on modern limit theory as a “strict mathematical proven” truth and a unimpeachable theorem------it would be Great Zeno’s Achilles--Turtle Race Theorem but not Suspended Zeno’s Achilles--Turtle Race Paradox.
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
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