Actual Infinity in Reality?

[GrandizerII]

There is no end, but no matter how far you go through the set, there is already an element to observe.

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Just use your imagination and assume all elements actually already exist in the set of positive integers:

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

The elements are all already there, including 11 and so on. Can you add more positive integers to the set that aren't already there yet? No! No matter how far you go through the set, any last integer you reach would have already been an element of the set. It didn't need your observation to bring it into being. Even if it was Graham's number, or TREE(3), they're already there (assuming they're integers, of course; if not, ignore this last sentence).

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.

Yes, "actual" means "complete" ... as opposed to "potential". It doesn't mean that, necessarily, an actual infinity must have ends. A complete set means that it has all the required elements in it (which is rather redundant to say, but this is what I have to deal with in order to explain this to you).