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I'm aware of such mathematical concepts as and potential infinity and actual infinity. But I do not understand how those concepts are being used. Are there any applications to such concepts? Are there different symbols used to denote potential infinity and actual infinity in order to avoid confusion? Maybe those types of infinities are rather philosophical concepts than mathematical? I would appreciate your thoughts on this matter.

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They are philosophical concepts, dating back to Classical Greece. – André Nicolas Oct 2 '12 at 19:20
@AndréNicolas Philosophy and mathematics are inextricably linked. I'm also interested in how ideas of potential and actual infinity enter into mathematics, or if they do not, why it's not a relevant distinction. – Richard Sullivan Oct 2 '12 at 19:27
Modern infinities come mainly from three areas: order theory (ordinal numbers), set theory (cardinal numbers) and geometry (points at infinity). The last concept is very different from the previous two. Greek math was not advanced enough to think about them separately, however, so I wouldn't pay a lot attention to these early crude ideas. – Alexei Averchenko Nov 2 '12 at 12:08
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I disagree with Hurkyl's answer, so I am making this an answer rather than a comment.

Peano's successor axiom says that for any integer $n$ we can construct its successor. This is a potential infinity, since we can get as large as we want (even to Graham's unimaginably large number). But we can never get all of the integers.

Extended to other parts of mathematics, we get constructive mathematics, where we can only talk about values we can explicitly construct (I am probably misrepresenting this, so please correct me as needed.).

In my opinion, there is a big (even infinite) difference between a potential infinity, where everything is explicitly constructable, and an actual infinity, where we consider all the items that can possibly be constructed.

As to which is true, let's take a vote!!!!

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I've spent a little bit of time trying to think through an ultrafinitist viewpoint: in the end, I think the point of "not getting all of the integers" is not what I would think ultrafinitistically. Rather than being an "internal" statement of how I would picture an ultrafinitist "universe", it is an external statement about the comparison between an ultrafinitist "universe" and a traditional "universe". In the end, it sort of felt like non-standard analysis in reverse, and reminded me a great deal of the philosophy often attached to internal set theory. – Hurkyl Oct 3 '12 at 0:17
To phrase that better, the integers we "can't" get are leftovers from prior, non-ultrafinitist thinking. If we can't get to them, should we really be carrying around the idea that they are there, but unreachable? Or should we be carrying around the idea that the ones we can get to are all there are? Or, at least, all we should be worrying about? – Hurkyl Oct 3 '12 at 0:58

I take the bold opinion that there is no meaningful difference: the distinction between "potential infinity" and "actual infinity" is outdated and boil down to a matter of abstraction

The most common example of potential infinity given is the distinction between the notions

  • Start with 0. Repeatedly add 1 to produce new numbers.
  • The collection of natural numbers.

The former is described as a "process" -- a method that can produce more and more numbers of greater and greater size without bound, but at each point during the iteration, only finitely many numbers of finite size have been produced. The enumeration is always "incomplete": the infinite is merely potential.

This is distinguished from the latter which is one mathematical object that has all of the natural numbers in it. It is "complete".

But is this true? Is there somehow "more" to the latter than the former? No! In fact, quite the opposite is true.

It is a common theme in mathematics that when you have some notion, you can work with that notion by finding ways to encode enough properties of that object to pick it out and work with it.

For example, if we want to work with a natural number, we might specify it by writing a list of its decimal digits. Or we might write an arithmetic expression that computes it. Or we might write an equation to which it is the unique solution.

If I have the abstract notion of "rational numbers", I might determine that any rational number can be completely determined as the ratio of two integers. And so working with ratios of integers becomes one of my tools for working with the concept of rational number.

Now, if I have the notion of a collection of objects, how might I describe such a thing? The two most 'natural' ways to do so are:

  • As a process that enumerates the objects in the collection
  • As a rule for answering the question of whether an object is in the collection

So returning to the original example, if I have the notion of "the (complete) collection of all natural numbers" in mind, the rule:

  • Start with 0. Repeatedly add 1 to produce new numbers.

is one of the most natural ways to specify this collection. In fact, the most standard way (Peano's axioms) to specify the collection of natural numbers is, at its heart, consists of nothing more than this process, and a formalization of the idea that every natural number arises from this process.

But not only does this process completely and unambiguously specify the entirety of the natural numbers, it also arranges them into a sequence, provides a recursive function that generates the sequence, and an algorithm for computing that function.

This additional information is above and beyond what the notion of the "(completed) collection of natural numbers" was trying to capture. Really, the difference between potential infinity and actual infinity, now, are seen to be the failure to take the notion of an "enumeration" and extract the abstract notion of "what is being enumerated".

When usual mathematics wants to talk about these additional ideas, it will use other objects, such as sequences, recursive definitions, or computable functions. We have much more precise means to talk about these ideas -- the notion of potential infinity is, IMO, an out-dated one.

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