Actual Infinity in Reality?

[SteveII]

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.

Yes, of course you need sets for addition (for example). Without them, how do you even define addition? Try to define the notion of addition without using collections. You cannot do it. The same goes for multiplication. Subtraction and division require *extensions* of those notions and the way to extend them is by careful consideration of the sets involved.

So, a challenge: how do you *define* addition? Use *only* those logical axioms that you have accepted prior to this. They are just not sufficient to the task. For that matter, even to define the concept of number requires the use of collections.

And *you* were the one wanting to get results like infinity+infinity=infinity from moving people around. YOU were the one claiming subtraction needs to be well defined for infinite quantities (with no reason).

You don't need sets for addition (or any other basic arithmetic operation). If you think so, it should be easy for a math student to post a link that explain it. Until I see an article from a recognized source of such things, I will assume you are wrong.
What is your fascination with what axioms I am using? It does not really matter, because we are not arguing about two logic systems. We are arguing using propositional logic. If you think that is the wrong logic system to use, make an argument why. Until then, stop thinking this is a point that needs to be made. Regarding addition and multiplication:

The operations of arithmetic on real numbers are subject to a number of basic rules, called axioms. These include axioms of addition, multiplication, distributivity, and order.
http://science.jrank.org/pages/505/Arith...z58VfUtp8b

Lest you think there is an argument to be made that HH example doesn't contain real numbers, the rooms are numbered--with real numbers.