1) By actual infinity I mean that given X, it is inaccessible by Y, where Y is a placeholder for any possible non-finite set, such that no non-finite set is accessible to X (X is beyond the notion of collection of objects, where set is a particular case of collection).

2) By (1) any given non-finite set it is permanently under construction (it can't have an accurate cardinal number), which means that Y is defined in terms of potential infinity with respect to X.

3) Both X and Y are used in the considered mathematical framework.

In order to be clearer I wish to share some of my notions about this question.

Here is a quote taken from Wikipedia:

"A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This means that one cannot "reach" λ from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear."

By Cantorian mathematical framework, actual infinity is (at least) defined in terms of sets, such that |N| is the weak limit cardinal of the set of all natural numbers (where |N| is the smallest weak limit cardinal).

If one defines actual infinity by going beyond sets, then any non-finite cardinal number that is based on sets, is inaccessible to actual infinity (it can't measure actual infinity), and therefore it can't have an accurate value (being inaccessible means permanently under construction).

This is exactly what I do by defining a given circle (which its circumference > 0) as a non-composed object (it is not defined as a collection (or more specifically, set) of objects (set of points, in this case)).

So, x (the circle's circumference) > 0, where the division operation is used here to define the number of points on the circle (I use the term "on" in order to indicate that a given circle is not defined as a collection (or more specifically, set) of objects (set of points, in this case)).

Let us use the framework of modular arithmetic in order to examine the concept of cardinal numbers, which "are a generalization of the natural numbers used to measure the cardinality (size) of sets" (quoted from Wikipedia).

An example:

Let B be a point and let A be a non-composed circle, such that it is not defined as a set of points.

Let the circumference of the non-composed circle be x, such that x > 0.

Let the division operation be used here in order to denote the number of Bs on A (we are using here the term "on" in order to indicate that A is not defined as a set of B objects).

By this framework:

x/0 indicates that there are 0 Bs on A (A is not defined as a set of B objects).

x/1 indicates that there is 1 B on A (A is not defined as a set of B objects).

x/2 indicates that there are 2 Bs on A (A is not defined as a set of B objects).

x/3 indicates that there are 3 Bs on A (A is not defined as a set of B objects).

Etc. ad infinitum … (where A is not defined as a set of B objects).

By this framework:

a. x/0 is a valid mathematical expression, which indicates that A is non-composed (A is not defined as a set of B objects).

b. By a., No cardinal number > 0 of Bs on A, defines A as non-composed (exactly because A is not defined as a set of B objects).

c. By b., A is defined in terms of actual infinity, where any non-finite cardinal number is a measurement of potential infinity (which means that infinite sets are defined in terms of potential infinity).

d. By c., no non-finite set has an accurate cardinal number (by this framework non-finite sets are defined in terms of potential infinity), since it is inaccessible to A, which is defined in terms of actual infinity.

e. Such framework is non-Cantorian.

The usefulness of such framework:

It defines infinite sets (or more generally, infinite collections of objects) as open systems, which may lead to better understanding of non-entropic systems (for example, living systems can be researched by mathematical tools that do not define the size of non-finite sets by fixed values, as done in case of finite sets).

Moreover, by looking at A as a non-composed 1-dim string-like object, it is actually used to gather any number > 0 of Bs into collections (of 0-dim beads-like objects), so we get a simple non-Cantorian model of the concept of set, which is a particular case of the concept of collection (generally, any pair of (n,k) dims, such that n=non-negative integer and k=non-negative integer > n, can be used (which means that modular arithmetic framework is some particular case of the considered mathematical framework)).

  • 2
    Calculus is an example if you consider its foundations which reside on set theory and recall that the limit operation and analysis encompass potential infinite. – user18096 Aug 20 '18 at 17:16
  • 4
    The title question is interesting, but most of the post does not help and should be deleted (most users would not read a wall of text). What "accessible" or " permanently under construction" mean in the opening lines is unclear and clarifying that might help. But if "actual" and "potential" are assigned some technical meanings then this is a technical question for Math SE or Math Overflow, not us. – Conifold Aug 21 '18 at 0:34
  • @Conifold, in order to know the meaning of "(in)accessible" or "permanently under construction", one needs to read what you call my "wall of text". If you actually read it, then it may be transformed from wall into a bridge to the considered subject. So please take your time for this transformation in your mind, if you still think that "The title question is interesting". Thank you. – doromshadmi Aug 21 '18 at 5:35
  • @Penguin, by axiomatic set theory actual infinity is based on the notion that non-finite number of objects are measured by an accurate value (transfinite cardinal number, for example |N|). My question is about a mathematical framework that goes beyond the notion of set in order to define actual infinity. Actually the aim of my question is to extend the understanding of potential/actual infinity beyond the notion of collection (where set is a particular case of collection). – doromshadmi Aug 21 '18 at 6:04
  • If you could prove mathematically that the infinity between 0 and 1 is the same entity as the infinity between 1 and infinity. You would prove that humans have no free will... Since you would be proving that the future already exists. There's a thought. – Richard Aug 21 '18 at 22:16

One way of looking at Nonstandard Analysis uses the distinction between actual and potential infinity to define 'internal' and 'external' objects.

For instance, if the Reals are defined as the points of convergence of sequences, the Non-Standard Reals are the convergence points of transfinitely long sequences, an idealized representation involving actual infinities and the Standard Reals are the convergence points of countably long sequences, a normal representation involving only potential infinities.

The injected ideal objects make arguments simpler, but a filter on the language determines which of those arguments predict true results about the universe without those idealized objects inserted.

This gives us a mathematical framework that makes use of the notion of actual infinities as a tool, without forcing us to consider anything beyond potential infinity to have any actual meaning.

  • 1
    So, in non-standard analysis, is there a formal distinction made between actual and potential infinity, e,g, a set is potentially/actually infinite if and only if... – Dan Christensen Aug 21 '18 at 4:33
  • Via the transfer principle, the standard and nonstandard models make exactly the same mathematical "predictions". – Hurkyl Aug 21 '18 at 6:47
  • Also, reasoning about the standard reals is exactly the same as reasoning about the nonstandard reals; if one is about "potential infinity" and the other "actual infinity", then the terms themselves have no mathematical content at all. (the power of nonstandard analysis is the additional methods of reasoning you gain by considering simultaneously both the standard and nonstandard models of real analysis) – Hurkyl Aug 21 '18 at 6:49
  • @DanChristensen if you read what I wrote about my question (which is inseparable of the question) you realize that it is not restricted to the concept of set. – doromshadmi Aug 21 '18 at 6:53
  • @Hurkyl, Standard or non-standard reals are based on the notion of collection. Actual infinity in my question is not restricted to this notion. So in order to value my question, please read what I wrote about it, before you vote about its value. In case that you have already voted, please explain your view by providing some details that may help to rest of the posters to learn something about your reasoning behind your vote. Thank you. – doromshadmi Aug 21 '18 at 7:09

Yes, there is a theory needing both notions, namely modern set theory.

It has finished or actual infinity, guaranteed by the axiom of infinity. There are all infinite sets actually infinite by the axiom of extensionality.

However there is no set of all sets. That means the hierarchy of sets is not complete. But since ordinal numbers and also cardinality have no upper bound, this hierarchy is potentially infinite.

  • if the notion of actual infinity is defined not in terms of composed forms like sets (for example, at least, ___ is axiomatically defined as a non-composed form), then the axiom of infinity of modern set theory does not guarantee sets (which composed forms) in terms of actual infinity. – doromshadmi Sep 17 '18 at 14:14

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