Actual Infinity in Reality?

[SteveII]

Why would we have to add all the rooms to a set? When we are counting and dividing up bricks to build four sides of a house, do we have to put them in a set to discuss them? What principle are you applying that requires it? Also, set theory is a specialized mathematical concept governed by a whole set of axioms. Why would you have to apply a mathematical concept to a simple accounting of rooms in a hotel? We wouldn't for a 100, 1000 or even 10^10 rooms.

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. 

You do not need sets to preform any basic operations like addition, subtraction, multiplication or division. I'll give you a chance to back away from that assertion. 

The only reason you desperately want to hang on to putting those rooms into sets is so you can apply unwarranted external constraints (set theory) to them and dismiss the contradiction and absurdities that would otherwise surface.