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Actual infinity, unlike potential infinity, is one interpretation of infinity where there is an end, of which we have no idea, say counting numbers (1,2,3,4,... it keeps going).

I have read about Cantor's ideas of countable and uncountable sets and his take on actual infinity.

My question is, and please correct me if I'm mistaken in my understanding, is whether a real interval (like (2,3)) an example of actual infinity. It clearly has a starting point and an end point. But, it also has infinitely many points with in it.

Asaf Karagila
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Sathvik R.
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    Depends how you're defining infinity. If by the 'length' of the set or the cardinality then you could get two different answers – Anonmath101 Jun 23 '21 at 10:49
  • It also (to follow up on @Anon) what you mean by a real interval. A single real interval is a single mathematical object, and no infinity seems to be involved if you don't view it as being made up of points (e.g. as in Euclid geometry). – Dave L. Renfro Jun 23 '21 at 11:01
  • @DaveL.Renfro and even if you see it as being built from points, if those points are limits of sequences, this can be seen as potential infinity. – Anixx Jun 23 '21 at 11:06

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The point of potential infinity is that it is only used as a reservoir for more stuff (numbers, points, functions, whatever) and that it will never end.

Actual infinity, in contrast, is when you have an actual mathematical object which is infinite.

People who reject actual infinity, but accept potential infinity, will tell you that the collection of all natural numbers is not an actual mathematical object. Sure, there are infinitely many of them, but the collection as a whole is not a concrete object.

On the other hand, in modern mathematics, almost any reasonable foundations of mathematics will have a statement which says that the collection of all natural numbers is an object (in the case of set theory, this is the Axiom of Infinity, for example).

The interval $(0,1)$ is an actual infinity, yes, in that it is a concrete object and it is infinite. But so is $\Bbb N$, the collection of natural numbers.

Asaf Karagila
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    I suppose people who reject actual infinity would say that the set of all real numbers between $2$ and $3$ is not an actual mathematical object, so in that sense the real interval $(2,3)$ does not exist. – David K Jun 23 '21 at 11:30
  • That is my understanding. It is a potential infinity, in the sense that you can find more and more points strictly between $2$ and $3$, but it's not a bona fida object. Of course, there is also the approach that an interval is not "made of points", but rather that it is a continuous object somehow. I find that approach to be quite flawed, though, so I rather ignore it altogether. – Asaf Karagila Jun 23 '21 at 11:34