Down to earth, a set whose number of elements grows over time to infinity I would call "potential infinite set":-)

Geng:

Some would say we only can think of potential infinity and cannot but postulate actual infinity.  Some would say (possibly Aristotle and Brouwer) that actual infinity does not exist and only the method of construction toward ever larger numbers is all we have.  Yet we use actual infinity in calculus - especially integral calculus where an infinite number of finite lines can be taken as a whole to measure the area of any plane figure.

Others, including Carl Boyer in his "The History of the Calculus and its Conceptual Development" suggest that a compromise was made between two feuding camps - from which we get the epsilon-delta definition of a limit.  The disputes around infinity - potential and actual - have been at the foundation of calculus and discussed over 300-400 years, and likely with only a compromise truce as a result.

The basis of the problem might be that actual infinity can only be handled as a concept without any verifiable examples.  Potential infinity can allow demonstrable proofs and examples.  The success of the integral and differential calculus indicates that the concept of actual infinity is, at the very least, extremely useful.  The unverifiable character of actual infinity would indicate a certain level of explicit 'faith' is required in the application of the calculus.

It seems to me that, given the usefulness of calculus, along with the unverifiable nature of actual infinity, that we cannot expect to resolve this issue using strict logic and proof (using our current tools of logic).

We might only be able to identify when we are working with one or the other - and apply rules appropriate to which topic is under consideration.  In this sense, the direction is to understand when we are dealing with one or the other so as not to apply the wrong rules to a situation (eg. applying 'potentially infinite' rules to an 'actually infinite' situation).  I am of the opinion that Cantor was not always clear about which situation he was handling and that he ended up - in certain places - confusing the two.  I am also not sure a clear separation is even possible, especially given the confusion between numbers and representations of numbers when used for 1-1 matching of infinite sets.

So, to question 1) I would say that only if the definitions of potentially infinite and actually infinite are adequately proscribed, might we be able not to confuse and 'jump between' these concepts.

A response to 2) is similar - in that unless adequate definitions, with suitable examples for clarification, exist, we cannot tell which 'side' we are dealing with.

Unless I mis-understand your definitions, an example of a 'potentially infinite' set seems easy - 1, 1+1, 1+1+1, ...  It is an example of an actually infinite set that seems the harder to define (eg. how to define 'the set of all Real numbers', except via constructive or 'potentially infinite' rules).

Arno:

Do you think there are many people who understand the nature of infinity?  I would question that any finite human can truly understand the nature of infinity...

Dear Colleagues, thank you for your frank points of view.

Dear Mr. Jerzy Hanckowiak, would you please show us the exact "potential infinite set" you have in your mind?

Dear Mr. Donald G Palmer, the followings are my points of view:

1, the endless and suspended debates of “what is potential infinite--what is actual infinite” prove that it is impossible to have adequate definitions to “potential infinite and actual infinite” in present classical infinite related science system. There will sure be a revolution to dispel the “2500 years old huge black cloud of infinite related paradoxes over mathematics sky” produced by the defects in present classical “potential infinite and actual infinite” related science system.

2, I agree with you that “It seems to me that, given the usefulness of calculus, along with the unverifiable nature of actual infinity, that we cannot expect to resolve this issue using strict logic and proof (using our current tools of logic)”. I think people have down a lot in the field of “practical infinite” but not enough in the field of “theoretical infinite”; so on the one hand, we have wonderful Mathematical Analysis, on the other hand, we have suspended infinite related paradoxes families since Zeon’s time and the modern version of Ancient Zeno’s Achilles--Turtle Race Paradox:

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                  （1）

=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．      （２）

>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．           （3）

=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                           （4）

 “Can we really produce infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 10000000000)”?.

3, if there is a definition for the set of 1, 1+1, 1+1+1, ... people call this set “an actually infinite set under a clear definition (within the boundary of the very definition)”.

Dear Mr. Arno Gorgels, very happy to meet you again and thank you for your frank points of view.

Best New year Wishes,

Geng

Just a note that there is a geometric sense of 'points at infinity' in the affine plane.  This are actual - and can be used to represent real finite parts of mechanical engineering linkages (sliders).  The connections are best made by looking at points on the sphere and their projections to the plane (from the center).  In that geometric process, the points on the equator project to 'points at infinity'.   

This is not the set theory sense of infinities and counting - but something which can be both approached in a continuous way and played with in algebra (projective coordinates) and with spatial visual reasoning. 

The debate on foundations, effectively eliminating the epsilon delta fix to analysis was resolved technically through developments in model theory leading to Robinsons Non standard analysis that looked back to Leibniz formulation while in corporating the axiomatic distinctions around which Russell and his contemporaries debated. It means that you can do mathematics constructively or intuitionistically or ignoring the wierdness of cantors infinities in each case to your own satisfaction. Nonstandard analysis simply rescues the calculus while providing an intriguing way to reformulate classical proofs without straining the bounds of  obse4vable or constructive sense. 

Dear Colleagues, thank you for your frank points of view.

Dear Mr. Walter Whiteley, Mr. Stephen Robert Horn and Mr. Arno Gorgels, thank you for your frank points of view.

1, do you agree that if there is a definition for a set, people call this set “an actually infinite set under a clear definition (within the boundary of the very definition)”?

2, according to my studies, Robinson’s Non Standard Analysis can do nothing to solve the defects disclosed by Zeon’s paradoxes families, especially the modern version of Ancient Zeno’s Achilles--Turtle Race Paradox-----the newly discovered Harmonic Series Paradox.

Happy New Year!

Sincerely, 

Geng

Dear Mr. Waldemar Koczkodaj, thank you for your interesting idea.

I think infinite can be divided into theoretical infinite and applied infinite. But finite, potential infinite, actual infinite are unavoidable confused and mixed up in present classical infinite related science system.

Happy New Year!

Sincerely, 

Geng

If we work in a set theory without the Axiom of Infinity, there are natural numbers, but no totality of natural numbers.  AI makes the distinction between "actual infinity" and "potential infinity" disappear. 

Dear Professor Paul Bankston, Professor Arno Gorgels and Professor Waldemar Koczkodaj,

It is believed that there is the bijection relationship between Rational Number Set and Natural Number Set, but following simple “story” tells us that Rational Number Set has far more elements than that of Natural Number Set:

Just take a tiny portion of rational numbers (elements) from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) and they map very well to all the numbers (elements) in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so a lot of rational numbers (elements) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set” and Rational Number Set is bigger than Natural Number Set. 

 What’s wrong------- Rational Number Set is uncountable?

Best New Year Wishes,

Geng

I also thought on the problem that  Professor Geng provided. I think that the known bijection between rational number set  and  natural number set is not correct. Thebijection between two infinite sets should have a clear formula. For example, the traditional bijection cannot tell us the rational number n/m corresponds to which natural number.

Hi Geng,  I agree with your 'story," but one can also map--say--the even natural numbers onto the entire set of rational numbers; thereby proving there are "more" natural numbers than there are rationals.  What I was saying before about the Axiom of Infinity has the following point:  Only by specifying what you mean by "set" can you make any sense of what "infinite" means. So, say, with ZF set theory, you can produce sets 0 (=empty set), 1 (={0}), 2 (={0,{0})), etc., the finite von Neumann ordinals, where n+1 is defined to be the union of n and {n}.  What AI postulates is that these finite ordinals may be packaged up into {0,1,2,...}, an object in our universe of discourse. In ZF minus AI, you can produce ever larger finite sets, but there is only "potential" infinity:  infinite objects do not exist as objects to be talked about. 

Dear colleagues, thank you for your frank points of view. Followings are my ideas:

1, the very “simple mapping story” between rational number set and natural number set is sure to happen between rational number set and even natural number set.

2, “observer's position” should be equal to every one in mathematics.

3,  there must be something wrong in the foundation of present classical infinite related science system, but what?

Best New Year Wishes,

Geng

Dear Arno， do you mean different human observer has different mathematics.

Dear Arno，according to your idea, we human have millions and millions of different mathematics, which mathematics you choose and …?

Yours,

Geng

Dear Arno，can we human have millions and millions definitions of infinite or only one definition of infinite in our science?

Yours,

Geng

Dear Arno，in my opinion, it is the time for us human to integrate those confusing and “trouble making” different definitions of infinite (such as "actual infinite" and "potential infinite") in our science ---------the task basing on the achievements people have done in our thousand years science history and standing on the giants shoulders.

We are ready.

Yours,

Geng

Dear Geng,

The debate over potential and actual infinities has been ongoing for many (NEARLY INFINITY) centuries., It’s worth to note that Cantor’s work on set theory has showed that the concept of an infinite set makes mathematical sense and avoids contradiction.  We are talking now about potential infinity, and infinity is how many you want, and even more. So it is not a number but an abstract concept dressed in the well known symbol. As you know, this infinty has different sizes (natural numbers are countable, real numbers are uncountable). Furthermore, there are infinity infinities.

To be short, according to me, potential infinity belongs to epistemology, and actuaal infinity belongs to ontology. The union of these two notions is possible in platonian world.

Regards, Tadeusz

Dear Professor Arno Gorgels and Professor Tadeusz Ostrowski，thank you.

Our science history tells us that for more than 2500 years, people have been “jumping up and down (on--off) as one wishes on the confusing actual infinite and potential infinite” and the “huge black cloud of infinite related paradoxes over mathematics sky” is produced by those defects disclosed by suspended Zeon’s Paradoxes family in present classical infinite related science system.

A revolution (in “infinite”, “infinite related number system”, “infinite related limit theory”) is on the way now.

In new infinite theory system, infinite is divided into “theoretical infinite and practical infinite” but not “potential infinite and actual infinite” and there will be no debates over potential and actual infinities any more.

Dear colleagues, anyone knows how to solve following problem:

The elements of tiny portion of rational numbers from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …)  map and use up all the numbers (elements) in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so infinite rational numbers (at least 2,3,4,5,6,…n,…) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set”------- Rational Number Set and Natural Number Set are not bijective at all.

What decides the same cardinality of two sets?

Regards,

Geng

Dear Geng, your map that "The elements of tiny portion of rational numbers from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …)  map and use up all the numbers (elements) in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …)" indicate that the cardinality of natural number set is less than or equal to the cardinality of real number set, according to the definition of Kantor. 

I think that the key problem is that what is the definition of the bijection between two infinite sets.

Potential infinity is just a matter of definition. The "set of numbers citizen X ever thought about" was all the time finite, but slightly increasing. As long as citizen X is alive, this set will be finite, but potentially infinite, because every second citizen X can think about a new number, he never thought about before. When citizen X will die, the set will remain finite and will be no more potentially infinite...

Dear Yancai: there may be many bijective ways between Rational Number Set and Natural Number Set but can one choose any way he likes freely as mathematical basic theory? Why?

Dear Mihai: the trouble is, in present classical infinite related science system with confusing “potential infinite” and “actual infinite” as its foundation, many people (including Kantor) really think we can and must “automatically” switch between “potential infinite” and “actual infinite” in practical infinite related mathematical operations.

Thank you!

Yous,

Geng

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