Actual Infinity in Reality?

[RoadRunner79]

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).

---

I'm sorry, I'm going to have to flunk you for not following directions (this is not common core).  You need to follow what Zeno had described.   And you do not reach the end of 1 (which is necessary) to be called infinite.  Infinitely small times still do not help you.  It's not a matter of time. But I do find that you saying that more time, would not allow you to finish, that you need infinity less time.... That is funny.

For the last part, perhaps I worded it incorrectly.  What is your last point, before you reach your destination (1 or whatever the number is)?  And you can declare it infinite as much as you like. You are assuming your conclusion in your premise (as Steve pointed out before).

Well, we can turn Zeno's arguments around and show that it *is* possible to go through an infinite collection of things. For example, to go forward in time one second requires we go through the first half second, the next quarter second, the next eighth of a second, etc. Since we *do*, in fact, manage to get past the one second mark, we do, in fact, manage to go through that infinity. It *is* completed once we hit the second mark on our clocks.

And the point for the spatial distances is that each of those infinitely many distances is paired with one of those infinitely many times. Once those infinitely many times have passed (which they will!), we have also gone through the infinitely many distances.

Perhaps what is confusing you is that we can add an infinite number of positive quantities and get a finite answer. But that is trivial to see:

1/3 =.333333.... = 3 +.03 +.003 +.0004 +.00003 +....

This all shows two things: 1. Finishing an infinite process is possible if the time for each step decreases geometrically. 2. It is the fact that space and time are *both* infinite in the same way that allows for motion.

So you just add  "...."   and then you finished?   That you are finishing a process that you are claiming is endless is a contradiction.    And the math in your dichotomy addition will never add up to what you want it to.  That is why it is an infinite chain (until you stop anyway).

Also a distance infers that you have two points or a segment.   You cannot add an infinite number of distances, and end in a result period (again by the nature of claiming that it is infinite).  The only exception perhaps is perhaps if you are adding a 0 distance an infinite number of times.

I also think that you are confusing "showing" with "assuming".  In my job, we sometimes run into engineers, that while they may be book smart, are said, to have not had enough time in the field.  The don't understand that the abstraction is not equal to actuality.  Your concept may have a perfect sphere, but in actuality, we get as close to a perfect sphere as we can (or hopefully, at least what we need).  In reality, we don't deal with points of zero size, if they have zero size, then they are not really anything.  It does depend on what your abstraction represents.   Can you actually cut a thing into a perfect 1/3, can you have a perfect circle, can we know that they are?  How many of these abstracts refer to something in them that is zero size?  What I think that you are showing, is that these things are not infinite at all, but do come to completion.