Actual Infinity in Reality?

[GrandizerII]

Yes, of course you put them in sets: the collection of bricks for each side. You don't *have* to, but you can then discuss the bricks that will go to each side. That is a collection, i.e, a set. 

Arithmetic is a specialized mathematical subject governed by a whole set of axioms. To justify those axioms requires the use of collections, i.e, sets.

So, when you add, what are you doing? You are taking two collections (or whatever sizes) and putting them together into a single collection and counting the number of things in the new set. That is how addition is *defined*.

The same goes for multiplication: How do you define multiplication? You repeat a set of one size, one copy for each element in a different set. Then we look at how many total elements there are. So to define multiplication, you need some sort of set theory.

But, more, to show those operations are well defined (that they give the same result no matter how you rearrange things) is crucial and also depends on set theory.

So, yes, if you are even attempting to *define* 10^10, you will need the concepts of set theory.

Not only are you correct when you say "you don't *have* to" put the rooms in sets, but why would you? The desk clerk is not pulling out paper and using {G1, G2, G3...} to move the guests around. He is making changes with real rooms and real guests and does not have to resort to creating abstract objects by grouping them together.

Because no matter how you put it, you are grouping things together in your favorite analogy. And you represent that mathematically with a set, just as you represent mathematically you grabbing 5 sticks from here and then grabbing 5 more sticks from them and counting them with 5 + 5 = 10.