Actual Infinity in Reality?

[polymath257]

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.