We are human beings, therefore, we can name anything by many names that suit us, or that put us apart from the reality. This we do in the hope to define reality in a better way (at least in our mind). We are also great in thinking we are great. The all knowing (a misnomer) scientists think they know everything, however, we also agree that the more we know, the more we come to know (in our hearts) how less we know ....

And, that is the beginning of a non-ending infinite. Even if the infinite is present around us (and for us) as a finite entity, we will never know it as we cannot get out of it. The one finite that is our home makes it infinitely impossible for its constituents to get out of it and survive to watch it (the finite).

Your question is cool, but could be a little warmer, or much cooler (just joking). It remains a matter of semantics for the linguists and communication experts to throw some light on it. We the scientists will try to go deeper once we define infinite, be that in a universal sense, or in a limited sense of the flight of our mind.

Hope to get some good discussion/dialogue on this question.

Thank you for your question and initial thoughts.

Regards

Suresh

   Classically the potentials are mathematical objects defined in front of a differential operator and therefore they are  always undetermined. Thus if you want to assign them one value you need to choose a certain extra condition. But this is not at all related with infinite.

   It is true that the Coulombic potential depends of the inverse of the distance to electric source that produce it. This means that when you approach to

    radius---> zero

   then the (for instance, scalar potential) tends to infinity. This is obviously a problem, but it is well solved with the renormalization of the charge or the vacuum in quantum field theory. The idea is that when you increase the potential approaching the charge, there is a moment that the energy is equal to the twice the mass of the electron (assumed source of the potentia in this case) and therefore it is possible to create electron-positron pairs (vacuum polarization) which decrease its value make it finite.

   I hope that this can helps.

Dear Dr. Suresh Vatsyayann and Dr. Daniel Baldomir, thank you for your insightful points of view.

Theoretically, at lest since Zeno’s time 2500 yeas ago, we human have been trapped deeply in an irresistible vortex of “confusing abstract concept of infinite (potential infinite) and its carriers (actual infinite)”-------such as the confusion of “abstract concept of fruit and its carriers of banana, apple, grape, …”. This fundamental defect disclosed by the old and new members of Zeno’s Paradox Families has been producing great ill effects on our quantitative cognizing activities to “infinite things” and endless the “potential infinite--actual infinite” debates turn to become “infinite--finite” debates at last.

In our practical operations, on the other hand, to many mathematicians, “the infinite related number forms are invisible and untouchable variables, they are not numbers” although they really see them, touch them and treat them as numbers very often (potential infinite or actual infinite?) while to many people who are working in practical science (such as physicists), they are brave enough to say “the infinite related number forms are visible and touchable numbers” (potential infinite or actual infinite?).

Best regards,

Geng

Sorry, perhaps I did not understood properly your question. I have answered you as physicist thinking in electric potential and its singularities. This was a problem because in Physics always the word infinity is a feature of a problem and it was developed all a theory (renormalization) to avoid it.

    On the other hand, in Mathematics always arises the concept of infinity in many different forms: limits which are the basics of the infinitesimal calculus, alephs (set theory), singularities (topology), etc...And curiously this concept never could be implemented in Physics for describing any theory, even in huge quantities as the Astrophysics employs.

    If you have in mind these all different aspects or applications of the infinity concept, perhaps you can define it in a different form, but at the end all must be equivalent: a subset has the same number as the whole set as their cardinality.

potential infinite may be related with  some range of values while actual infinite is related with exact values. Potential infinite may be related with either it has  end after some stage or  no end  in any stage.But actual infinite will show it stands  to be end at  some specific position .

Dear Geng,

joking with words, I should say that your question is “infinite”!

Within a reasonable space that we can use in the RG Section for Questions and Answers, I shall limit my considerations and have chosen the topic: ‘Cantor and the transfinite’.

Cantor declares to be a misuse the potential infinite (the Greek ‘apeiron’) as it relates to a finite magnitude, albeit variable, and underlines its dependence from current infinite. This, however, understood as eternal and uncreated, is entirely unknowable by the human mind. The infinite that can be the object of knowledge is - according to Cantor - the ‘transfinite’, as 'created' and capable of being increased.

To persuade of the existence of ‘transfinite’ are the transfinite numbers of which Cantor, in line with the definition of ‘number as a set’ supplied by Frege and Dedekind, illustrates the construction criteria. In the wake of Galileo and Leibniz, Dedekind had pointed out a paradox of some infinite sets: the fact that their elements can be put into bijective mapping with the elements of a subset of them, contradicting the traditional idea that the whole is greater than a part thereof.

Dedekind and Cantor make of this aspect a point of strength as a basis to define an infinite set, and reserving finite sets the characteristic that if B is a proper subset of A, then the number of elements in A is greater than the number of elements in B. Thus Cantor, moving from cardinality and ordinality concepts, presents an ‘infinite method’ to measure the size of infinite sets and a method to order the elements in succession. Then, he goes on to define the concept of cardinal and ordinal number; he constructs a succession for size of infinite sets, each set associated with a transfinite number (cardinal or ordinal), and shows that there is no greater cardinal and ordinal number: the construction of transfinite numbers goes on endlessly.

As a conclusion of this brief note, we recognize that Cantor admitted the existence of an "infinity of infinities", the so-called absolute infinite, unknowable and unreachable at the height of the "tower of the infinite" that he himself had contributed to raise, fighting against the too backward ideas and men compared to the innovative capacity of this great mathematician.

Dear Dr. Suresh Vatsyayann, thank you.

I deeply understand what you say in above post. The papers I have up loaded on RG, especially the latest one  On the Quantitative Cognitions to “Infinite Things” (I)  express “different aspects or applications of the infinity concept” in my mind.

Best regards,

Geng

Dear Dr. Manoj Kumar Srivastav, thank you.

The problem is there are so far no clear definitions for “potential infinite” and “actual infinite” in our present classical infinite related science and mathematics theory systems. So, this fundamental defect disclosed by the old and new members of Zeno’s Paradox Families has been producing great ill effects on our quantitative cognizing activities to “infinite things” and endless “potential infinite--actual infinite” debates. Many people really use “may be” to talk about and have their operations on “potential infinite--actual infinite” and this “may be” causes many troubles------the newly discovered Harmonic Series Paradox is a most typical example.

Best regards,

Geng

Dear Dr. Gianrocco Tucci, thank you for your frank point of view.

Many people really believe that we can have as many definitions for “infinite” as we like and play with the concept of “infinite” very much freely-------creating the concepts of “infinite”, “super infinite”, “super super infinite”, “super super super infinite”,… by cutting an “infinite thing” again and again with “the first generating principle”, “the second generating principle”, “the third generating principle”, …; Infinite Set A has more infinite elements than those in the Infinite Set B, “more infinite”, “more more infinite”, “more more more infinite”,…; in our daily life, a well known story of Hilbert’s Infinite Room Hotel tells its lodgers: although there are no vacancy in the hotel (already one—to—one bijection between infinite lodgers and infinite rooms), there are still “more infinite rooms”, “more more infinite rooms”, “more more more infinite rooms”,… for lodgers in the hotel.

…

Best regards,

Geng

Dear Dr. Claus Janew, thank you.

It is true that there are “potential infinite” and “actual infinite” in our present infinite related classical science theory system at least since Zeno’s time, But the problem is the confused “potential infinite--actual infinite” have being creating a lot troubles (such as infinite related paradoxes) when we are treating (cognizing) some “infinite things” in mathematics. The Harmonic Series Paradox is a typical example:

1, no one knows what Un--->0 in Harmonic Series means-------“potential infinite” or “actual infinite”,

2, no one knows what differences between Un--->0 in Harmonic Series and dx--->0 in calculus,

3, no one knows whether or not we can really produce (with present classical  infinite theory) infinite constants each bigger than 1/2, or 100, or 1000000, or 10000000000000000000, or … from the Un--->0 Harmonic Series.

 Best regards,

Geng

Could any people tell what “those items of Un--->0 in Harmonic Series for producing (with present classical infinite theory) infinite constants each bigger than 1/2, or 100, or 1000000, or 10000000000000000000, or…” are-------“potential infinite” or “actual infinite”?

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