Actual Infinity in Reality?

[polymath257]

What are you saying that the term “actual” means in the term.  
I take it to mean that it is completed or actualized.  Yet the term infinite means without limit or end... or in other words never completed. It is similar to a square circle.
There also would not be a total quantity of the set.

OK, so part of your problem is your definition of the term 'infinite' as being without limit or end. That is NOT the best definition of the term and is not the one used for infinite sets.

For collections of objects (sets), the term infinite means 'not finite'. In turn, 'finite' means that it can be put into correspondence with a positive integer.

So, the set {1,5,8} can be put into correspondence with the positive integer 3, so it is a finite set of size 3. The set {1,4,9,16,25} can be put into correspondence with the number 5 and so it is a set of size 5.

Being infinite just means there is no positive integer that can be put into correspondence with that set. So, for example, the set of ALL positive integers, call it N, cannot. So it is an infinite set. It is a *completed* set because we know exactly which things are in it and which are not. So there is no ambiguity in the set, no process going on that is generating the set. It is just there.

Now, we often write this set in the following way,

N={1,2,3,4,5,...}

but this is a convenience based on our understanding that the ellipses represent a pattern we can use to determine what is in and what is not in that set. It isn't a process, but a pattern.

Now, quantities are a bit strange when it comes to infinite sets. We say two sets have the same 'cardinality' when they can be put into correspondence with each other. A classical example due to Galileo (mentioned by Steve) is to look at the collection of perfect squares

S={1,4,9,16,25,36,....}

and compare it to the positive integers

N={1,2,3,4,5,6,....}

Every square can be paired off with a positive integer:

1 <-> 1
4 <-> 2
9 <-< 3
16 <-> 4
..
..

Again, both sets are *complete*; we know exactly which things are in both. The pairing is simple, but is even easier backwards: each integer x is paired with its square x*x. This gives a correspondence between the two sets. Because of this, we say the two sets have the same cardinality (loosely, they have the same size).

Now, you may have noticed that S is a subset of N: everything in S is also in N. And it is a proper subset: there are things in N that are not in S. For finite sets, a situation like this would force the size of S to be smaller than the size of N. For infinite sets, all we get is that S is no larger than N. And, in this case, they have the same 'size' in the sense of cardinality. That isn't a contradiction, it is simply a way in which finite and infinite sets act in different ways. In fact Dedekind defined an infinite set to be one that can be put into correspondence with a proper subset. For him, finite means no such correspondence is possible.

So the set S can be put into correspondence with N, the set of positive integers. Any infinite set for which this can be done is called countably infinite. Yes, I know you can't count all the elements, but the idea of the terminology is that we can still use 'numbers' to 'count' them in some way. In any case, it is now standard terminology.

Now, it turns out that there are infinite sets that *cannot* be put into correspondence with N. We say those sets are uncountable. The collection of all real (i.e, decimal) numbers is an uncountable set. So is the collection of all subsets of N.

In practice, physicists use the collection of real numbers all the time. Itis the realm of calculus. It is the realm of continuous functions and integrals, of differential equations and motion. The uncountably infinite set of real numbers is used as a matter of course in all of physics. So, all modern physics is based on the usage of not just actual infinities, but actual uncountably infinite sets.

Since calculus is foundational in physics and uncountable sets are foundational for calculus, the whole notion of objecting to actual infinities simply is silly. The issue literally never arises.

In fact, anyone using the terms 'potential infinity' today is showing they are basing their ideas on concepts that have been discarded. NOBODY uses these Aristotelian concepts *except* philosophers. Everyone else has progressed past them. NO modern math or physics even uses the term.