Actual Infinity in Reality?

[RoadRunner79]

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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.

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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.

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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).