RE: Actual Infinity in Reality?
March 1, 2018 at 7:45 am
(This post was last modified: March 1, 2018 at 8:06 am by polymath257.)
(February 28, 2018 at 9:03 pm)SteveII Wrote:(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.
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).
(February 28, 2018 at 7:39 pm)RoadRunner79 Wrote:(February 28, 2018 at 6:41 pm)polymath257 Wrote:
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).
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.
