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February 28, 2018 at 4:45 pm (This post was last modified: February 28, 2018 at 5:03 pm by SteveII.)
(February 27, 2018 at 11:58 pm)polymath257 Wrote: I bring in set theory because you have postulated an infinite collection of rooms. That collection of rooms *is a set*. That's all that sets are: collections of objects. In particular, the collection of rooms in the HH is an infinite set. Because of this, already adopted the axiom of infinity when you start asking these questions. So, it was not I that begged the question, but you.
And right there is the problem with ALL of your reasoning to date on this subject. An infinite number of rooms is not a set. The rooms are rooms. Distinct objects in their own right. There is no justification to collect the rooms into some mathematical abstract object and treat them as a block with a different set of rules. Sets only exist on paper--in mathematics.
Quote:In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. [opening sentence in the article: https://en.wikipedia.org/wiki/Set_(mathematics)] emphasis added
(February 27, 2018 at 11:58 pm)polymath257 Wrote: I bring in set theory because you have postulated an infinite collection of rooms. That collection of rooms *is a set*. That's all that sets are: collections of objects. In particular, the collection of rooms in the HH is an infinite set. Because of this, already adopted the axiom of infinity when you start asking these questions. So, it was not I that begged the question, but you.
And right there is the problem with ALL of your reasoning to date on this subject. An infinite number of rooms is not as set. The rooms are rooms. Distinct objects in their own right. There is no justification to collect the rooms into some mathematical abstract object and treat them as a block with a different set of rules. Sets only exist on paper--in mathematics.
Quote:In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. [opening sentence in the article: https://en.wikipedia.org/wiki/Set_(mathematics)] emphasis added
So we cannot look at the collection of all the rooms? Why not? Whether or not it is 'mathematical', does it make sense to talk about the collection of rooms?
February 28, 2018 at 5:00 pm (This post was last modified: February 28, 2018 at 5:04 pm by SteveII.)
(February 28, 2018 at 4:48 pm)polymath257 Wrote:
(February 28, 2018 at 4:45 pm)SteveII Wrote: And right there is the problem with ALL of your reasoning to date on this subject. An infinite number of rooms is not a set. The rooms are rooms. Distinct objects in their own right. There is no justification to collect the rooms into some mathematical abstract object and treat them as a block with a different set of rules. Sets only exist on paper--in mathematics.
So we cannot look at the collection of all the rooms? Why not? Whether or not it is 'mathematical', does it make sense to talk about the collection of rooms?
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.
(February 27, 2018 at 10:44 pm)RoadRunner79 Wrote: For Zeno's paradox, it was offered that calculus is the answer, and that a infinite small amount of time, and an infinitely small distance can be covered. I don't buy it. It doesn't matter if you have a lot of time, or infinitely small amounts of time, or however much time. If you follow that procedure of infinitely dividing in half the remainder, you will never reach the destination. It's not a matter of time. This may also be demonstrating the issue above of using infinity as a number. Also, you cannot have an infinite number of small distances in a finite distance, nor an infinite amount of time, in a finite time. This is a contradiction. Zeno in his paradox, was arguing that there was no motion. And we can easily observe that his conclusion is false. Not only can I set out to and reach a distance of 10 ft. I may very well pass it. Who are you going to believe "Zeno... or you lying eyes?" There is nothing wrong with Zeno's math, the simplest answer is that there is a problem with his underlying assumptions.
Let me give a brief treatment of Zeno's paradox.
Let's start with the one that says to move from point A to point B, you have to first reach the half-way point, then the half-way point to that, then the half-way point to that, etc.
The claim is that we cannot move through an infinite collection of points in a finite amount of time. So, what *actually* happens?
Suppose you move a total distance of 100 feet in a total time of 4 seconds. That means that the time it took to get to the half-way point, 50 feet, is half the time: 2 seconds. When you are at the one-quarter point, 25 feet, you are at the one quarter time: 1 second.
What you find is that every distance has a corresponding time associated with it. In fact, for any distance you mention (say 30 feet), I can give you the precise time you were at that point (30/4=7.5 seconds).
And we are guaranteed to go through those times! By whatever mechanism, we actually do go through the half-way time, the quarter-way time, etc. We do go through those times! And since each of those times is paired with a specific distance, we can *also* go through all those spatial points.
it is the very fact that both space and time have the same infinite size that makes motion possible.
We can do similar things with the other Zeno paradoxes. Take, for example, the paradox of Achilles and the Tortoise. They run a race where the Tortoise starts out 100 feet ahead. Achilles runs 10 times as fast, though. So we all know that Achilles will catch up and pass the Tortoise.
Zeno, however, argues as follows: when Achilles is at the place the Tortoise started, the Tortoise is 10 feet ahead. When Achilles runs that 10 extra feet, the Tortoise is 1 foot ahead. When Achilles runs that 1 foot, the Tortoise is still 1/10 of a foot ahead. Thereby Zeno claims that Achilles can never catch the Tortoise.
So, let's bring time into play and see how that affects things. Suppose for definiteness sake that Achilles runs 10 feet per second and the Tortoise 1 foot per second. When Achilles reaches the place where the Tortoise started, the Tortoise is, indeed 10 feet ahead. This has taken 10 seconds. When Achilles runs that 10 feet, the Tortoise is now 1 foot ahead and the clock has added a second to give 11 seconds since the start of the race. After another foot for Achilles, the Tortoise is 1/10 of a foot ahead and the clock reads 11.1 seconds.
What we realize is that the distance Achilles has gone is getting closer and closer to 111.1111... feet, the Tortoise has gone 11.111... feet, and the clock reads 11.1111.. seconds.
In other words, Achilles has gone 111 1/9 feet, the Tortoise has gone 11 1/9 feet and the clock has read 11 1/9 seconds. What we have found is the exactly instant that Achilles has passed the Tortoise! Far from showing Achilles cannot pass the Tortoise, this actually zooms in on the exact time and location where and when Achilles does so.
Again, the fact that time is just as divisible as space is what resolves the paradox and even gives an interesting way to find when the event of interest happens!
So, very far from being the cause of Zeno's paradoxes, infinite divisibility is exactly how those paradoxes are resolved! The fact that space and time are *equally* divisible is what allows motion to be possible. If one were finite and the other infinite, motion could not happen.
If both are finite, what we would find is that only very specific speeds would be allowed. But there is no reason to think we can't go half as fast as any speed we are currently going. So, if anything, this shows that an infinitely sub-dividable space and time is required for motion.
This, by the way, is a good reason to adopt at least infinite divisibility and thereby an actually infinite number of spatial points and time locations.
Thanks for taking the time for this. I think it is a very nice lesson in the elementary school problem of "a train leaves Chicago at 10AM, what time does it arrive in Boston". And yes, you can tell where the train is, by what time it is or vice versa (assuming a constant speed of course and not taking into account acceleration; which would require more).
However this doesn't address the dichotomy paradox by Zeno. It's not about motion, or answering how we get from position 0 to position 1. I wasn't as familiar with them, but I do see that some of the instances of the paradox do have an addition of saying finite time. The ones that I was familiar with did not have this mention of time. And really it is unnecessary. As I mentioned before, it does not matter if it is a small amount of time, a large amount of time, or an infinite amount of time. The method that Zeno described of cutting in half and then taking the remainder and cutting it in half, and then repeating this over and over; does describe an infinite process. Using this method, you will never be able to get to a value greater than or equal to 1 (or whatever you destination is). In programming, this would be an infinite loop, that never completes (or for as long as the processor can handle very small numbers). Zeno's math and the conclusion, that you could never reach the end are both correct. If your infinite series ends, then it is not infinite. Time does not factor into the problem.
In the problem of Achilles and the Tortoise, you will need time, because it concerns speed, but it is basically the same problem, and you seem to be missing what the issue is. Infinite divisibility is the problem, not the solution. It is not an issue, if there is a minimum movement, or a minimum speed. But crossing infinite spatial points (which I would point out are still undefined) does cause a problem. If you are saying that this does cross the boundary of the destination (which is wrong mathmatically). What is the answer to where is the point 1 step before where you are at or past the destination (1)? It still also seems that your points have nothing to do with distance. Otherwise you would have an infinite amount of distance and a finite distance, which would be contradictory.
It is said that an argument is what convinces reasonable men and a proof is what it takes to convince even an unreasonable man. - Alexander Vilenkin If I am shown my error, I will be the first to throw my books into the fire. - Martin Luther
February 28, 2018 at 6:41 pm (This post was last modified: February 28, 2018 at 6:49 pm by polymath257.)
(February 28, 2018 at 5:00 pm)SteveII Wrote:
(February 28, 2018 at 4:48 pm)polymath257 Wrote: So we cannot look at the collection of all the rooms? Why not? Whether or not it is 'mathematical', does it make sense to talk about the collection of rooms?
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.
(February 28, 2018 at 6:11 pm)RoadRunner79 Wrote:
(February 28, 2018 at 12:24 am)polymath257 Wrote:
Let me give a brief treatment of Zeno's paradox.
Let's start with the one that says to move from point A to point B, you have to first reach the half-way point, then the half-way point to that, then the half-way point to that, etc.
The claim is that we cannot move through an infinite collection of points in a finite amount of time. So, what *actually* happens?
Suppose you move a total distance of 100 feet in a total time of 4 seconds. That means that the time it took to get to the half-way point, 50 feet, is half the time: 2 seconds. When you are at the one-quarter point, 25 feet, you are at the one quarter time: 1 second.
What you find is that every distance has a corresponding time associated with it. In fact, for any distance you mention (say 30 feet), I can give you the precise time you were at that point (30/4=7.5 seconds).
And we are guaranteed to go through those times! By whatever mechanism, we actually do go through the half-way time, the quarter-way time, etc. We do go through those times! And since each of those times is paired with a specific distance, we can *also* go through all those spatial points.
it is the very fact that both space and time have the same infinite size that makes motion possible.
We can do similar things with the other Zeno paradoxes. Take, for example, the paradox of Achilles and the Tortoise. They run a race where the Tortoise starts out 100 feet ahead. Achilles runs 10 times as fast, though. So we all know that Achilles will catch up and pass the Tortoise.
Zeno, however, argues as follows: when Achilles is at the place the Tortoise started, the Tortoise is 10 feet ahead. When Achilles runs that 10 extra feet, the Tortoise is 1 foot ahead. When Achilles runs that 1 foot, the Tortoise is still 1/10 of a foot ahead. Thereby Zeno claims that Achilles can never catch the Tortoise.
So, let's bring time into play and see how that affects things. Suppose for definiteness sake that Achilles runs 10 feet per second and the Tortoise 1 foot per second. When Achilles reaches the place where the Tortoise started, the Tortoise is, indeed 10 feet ahead. This has taken 10 seconds. When Achilles runs that 10 feet, the Tortoise is now 1 foot ahead and the clock has added a second to give 11 seconds since the start of the race. After another foot for Achilles, the Tortoise is 1/10 of a foot ahead and the clock reads 11.1 seconds.
What we realize is that the distance Achilles has gone is getting closer and closer to 111.1111... feet, the Tortoise has gone 11.111... feet, and the clock reads 11.1111.. seconds.
In other words, Achilles has gone 111 1/9 feet, the Tortoise has gone 11 1/9 feet and the clock has read 11 1/9 seconds. What we have found is the exactly instant that Achilles has passed the Tortoise! Far from showing Achilles cannot pass the Tortoise, this actually zooms in on the exact time and location where and when Achilles does so.
Again, the fact that time is just as divisible as space is what resolves the paradox and even gives an interesting way to find when the event of interest happens!
So, very far from being the cause of Zeno's paradoxes, infinite divisibility is exactly how those paradoxes are resolved! The fact that space and time are *equally* divisible is what allows motion to be possible. If one were finite and the other infinite, motion could not happen.
If both are finite, what we would find is that only very specific speeds would be allowed. But there is no reason to think we can't go half as fast as any speed we are currently going. So, if anything, this shows that an infinitely sub-dividable space and time is required for motion.
This, by the way, is a good reason to adopt at least infinite divisibility and thereby an actually infinite number of spatial points and time locations.
Thanks for taking the time for this. I think it is a very nice lesson in the elementary school problem of "a train leaves Chicago at 10AM, what time does it arrive in Boston". And yes, you can tell where the train is, by what time it is or vice versa (assuming a constant speed of course and not taking into account acceleration; which would require more).
However this doesn't address the dichotomy paradox by Zeno. It's not about motion, or answering how we get from position 0 to position 1. I wasn't as familiar with them, but I do see that some of the instances of the paradox do have an addition of saying finite time. The ones that I was familiar with did not have this mention of time. And really it is unnecessary. As I mentioned before, it does not matter if it is a small amount of time, a large amount of time, or an infinite amount of time. The method that Zeno described of cutting in half and then taking the remainder and cutting it in half, and then repeating this over and over; does describe an infinite process. Using this method, you will never be able to get to a value greater than or equal to 1 (or whatever you destination is). In programming, this would be an infinite loop, that never completes (or for as long as the processor can handle very small numbers). Zeno's math and the conclusion, that you could never reach the end are both correct. If your infinite series ends, then it is not infinite. Time does not factor into the problem.
In the problem of Achilles and the Tortoise, you will need time, because it concerns speed, but it is basically the same problem, and you seem to be missing what the issue is. Infinite divisibility is the problem, not the solution. It is not an issue, if there is a minimum movement, or a minimum speed. But crossing infinite spatial points (which I would point out are still undefined) does cause a problem. If you are saying that this does cross the boundary of the destination (which is wrong mathmatically). What is the answer to where is the point 1 step before where you are at or past the destination (1)? It still also seems that your points have nothing to do with distance. Otherwise you would have an infinite amount of distance and a finite distance, which would be contradictory.
And if each step of the process took the same amount of time, you would never finish. But if it takes geometrically less time, then you will. In fact, we *do* go through an infinite sequence. So the basic assumption that this is impossible is just false. Space and time are continua and not discrete.
Yes, Achilles actually does catch up with the Tortoise in a finite amount of time. If you look at Zeno's paradox, you realize his fundamental assumption is that you cannot go through an infinite number of points. That is what is incorrect. Not only is it possible, but it is required for motion.
In your question of what happened one step before, there is some ambiguity. Achilles does not take a *step* at each stage of this process. In fact, the tail end of the process happens in the interval of a fraction of one of his steps.
And there isn't a 'step before' in this process. It is an infinite, completed process. As a function of time, the graph of the stages taken is not continuous, but the motion itself is. That just shows the stages aren't a good description.
(February 28, 2018 at 5:00 pm)SteveII Wrote: 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.
(February 28, 2018 at 6:11 pm)RoadRunner79 Wrote: Thanks for taking the time for this. I think it is a very nice lesson in the elementary school problem of "a train leaves Chicago at 10AM, what time does it arrive in Boston". And yes, you can tell where the train is, by what time it is or vice versa (assuming a constant speed of course and not taking into account acceleration; which would require more).
However this doesn't address the dichotomy paradox by Zeno. It's not about motion, or answering how we get from position 0 to position 1. I wasn't as familiar with them, but I do see that some of the instances of the paradox do have an addition of saying finite time. The ones that I was familiar with did not have this mention of time. And really it is unnecessary. As I mentioned before, it does not matter if it is a small amount of time, a large amount of time, or an infinite amount of time. The method that Zeno described of cutting in half and then taking the remainder and cutting it in half, and then repeating this over and over; does describe an infinite process. Using this method, you will never be able to get to a value greater than or equal to 1 (or whatever you destination is). In programming, this would be an infinite loop, that never completes (or for as long as the processor can handle very small numbers). Zeno's math and the conclusion, that you could never reach the end are both correct. If your infinite series ends, then it is not infinite. Time does not factor into the problem.
In the problem of Achilles and the Tortoise, you will need time, because it concerns speed, but it is basically the same problem, and you seem to be missing what the issue is. Infinite divisibility is the problem, not the solution. It is not an issue, if there is a minimum movement, or a minimum speed. But crossing infinite spatial points (which I would point out are still undefined) does cause a problem. If you are saying that this does cross the boundary of the destination (which is wrong mathmatically). What is the answer to where is the point 1 step before where you are at or past the destination (1)? It still also seems that your points have nothing to do with distance. Otherwise you would have an infinite amount of distance and a finite distance, which would be contradictory.
And if each step of the process took the same amount of time, you would never finish. But if it takes geometrically less time, then you will. In fact, we *do* go through an infinite sequence. So the basic assumption that this is impossible is just false. Space and time are continua and not discrete.
Yes, Achilles actually does catch up with the Tortoise in a finite amount of time. If you look at Zeno's paradox, you realize his fundamental assumption is that you cannot go through an infinite number of points. That is what is incorrect. Not only is it possible, but it is required for motion.
In your question of what happened one step before, there is some ambiguity. Achilles does not take a *step* at each stage of this process. In fact, the tail end of the process happens in the interval of a fraction of one of his steps.
And there isn't a 'step before' in this process. It is an infinite, completed process. As a function of time, the graph of the stages taken is not continuous, but the motion itself is. That just shows the stages aren't a good description.
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).
It is said that an argument is what convinces reasonable men and a proof is what it takes to convince even an unreasonable man. - Alexander Vilenkin If I am shown my error, I will be the first to throw my books into the fire. - Martin Luther
February 28, 2018 at 9:03 pm (This post was last modified: February 28, 2018 at 9:04 pm by SteveII.)
(February 28, 2018 at 6:41 pm)polymath257 Wrote:
(February 28, 2018 at 5:00 pm)SteveII Wrote: 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.
(February 28, 2018 at 5:38 am)robvalue Wrote: I answered yes. I see no reason why an infinite number of things can't logically exist. Whether there actually are an infinite number of things, I have no idea at all.
I think they're not actually physical possible so I voted no but there's no reason to say they're not logically possible, yes
February 28, 2018 at 10:03 pm (This post was last modified: February 28, 2018 at 10:54 pm by GrandizerII.)
(February 28, 2018 at 6:11 pm)RoadRunner79 Wrote: However this doesn't address the dichotomy paradox by Zeno. It's not about motion, or answering how we get from position 0 to position 1.
The whole point of Zeno's paradoxes is that they are arguments against motion, continuous or discrete.
I love, by the way, how Steve has not responded to my last response or two and is just repeating the same old bullshit that has long been debunked. So much for intellectual honesty.
(February 28, 2018 at 3:32 pm)SteveII Wrote: Since (2) is of the class of axioms that are not self-evident, they are assumptions on which further mathematical equations can be developed (useful in calculus for example). To be clear, this axiom is not reasoned into--it is just assumed as a foundation for the subset of infinite set theory in mathematics. See this earlier post.
This is CLEARLY a question-begging argument and therefore invalid.
That, or you still haven't grasped (or maybe you don't want to grasp) the exact argument we're making. We're not arguing that infinity is logically possible, therefore it is. We're arguing that you haven't provided any good reductio ad absurdum arguments to disprove the logic of actual infinity. You've failed with all the analogies and paradoxes you've posted about thus far.
And even if it was a question-begging argument, which it isn't, it would still be a valid argument anyway. Circular yes, invalid no.
And about the B-theory of time, William Lane Craig himself said the KCA is predicated on the A-theory of time and that if the B-theory of time is true, then the argument is useless. Well, we know which theory of time is backed by the science, and it's not the A-theory of time.
(February 28, 2018 at 6:41 pm)polymath257 Wrote: 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.
