Actual Infinity in Reality?

[polymath257]

1. The end (1) isn't reached *in that sequence*, but it is still reached. We get out of that sequence after a finite amount of time. Again, you assume we cannot go through an infinite collection of steps. We do because there are also infinitely many times.

2. We 'go through them all' by having, for each real number between 0 and 1 (inclusive, this time), a corresponding time when we go through it. And that is the case: for each real number, we have a corresponding time when we are at that real number. We also have a time when we at at 1. The fact that we go through an infinite number of times is what allows us to go through an infinite number of places. That is why motion is possible.

3. We have completed that infinity when we get to 1. And since there is a time when we get to 1, we have completed that infinity at that point.

I agree that there are an infinite number of points to go through. But I also think we do, in fact, go through them all. In fact, we have gone through all of them by the time we reach 1. Your assumption is that they cannot be crossed, when each and every one of them *is* crossed at some time.

1.)  So after a certain amount of time has passed, then we jump out of the infinite sequence?  Also, it's not my assumption, you agreed that the math with an infinite number of halves would never reach 1.  Do you still agree with this?

2.)  So if you have an infinite number of times (not infinite amount of time mind you),  then that will make the X in X<Y<1 be equal or greater than 1?   I think you need to be more specific on how this modifies what we previously talked about.  How does time effect X? 

3.)  You are just restating your conclusion as a premise.   The question was... where do you think the error is, in Zeno's statements and logic.   Do you disagree with the premises?   Do you disagree with the math?   Do you think that the conclusion does not follow? 

You may not like the definition of infinite to mean "without inherent limit or without end", however your definition is without end as well .   That is what it means to be infinite and while it make take on a different nuance when dealing with infinite sets, it does change this fact.  You seem to think that a sequence that doesn't end, does end.   And there is your contradiction.  Either it doesn't end and cannot reach the end or it does end, and is not infinite.  You can have it both ways.

Let's be clear about everything here.

1. What is the process we are using in going from x=0 to x=1?  Answer: We move with a speed of .2 units per second. In this, the time variable, t, goes from 0 to 5. For convenience, we can also assume we continue going past t=5, say to t=6 and thereby reach x=1.2.

2. What does it mean to say we go through every point? Answer: For each x, there is a t such that we are at x at time t.

Now, is it true that we go through every x between 0 and 1, inclusive? Answer yes. If we are curious what time we are at x, then t=5x will work.

So, is there a time for each point of the sequence of halves? Answer: Yes. In fact, for *every* x, there is a t, so we go through every point. So, in particular, for each x value in that sequence there is a corresponding t value when we pass through it.

Now to answer your questions specifically.

1. Do we 'jump out' of the sequence at some point? Actually, we jump in and out of the sequence many times. We are only in the sequence at times t=2.5, 3.75, 4.375,... ALL the rest of the times between t=0 and t=5, we are out of the sequence. So, for example, when t=2, we are at x=.4 which is not in the sequence. But, we are *past* that sequence at t=5, in which case, x=1. At no point *in the sequence* are we equal to, or greater than 1. But we get out of the sequence, none the less.

2. Since you assumed X<1, there is no time when X=1. But there is a *time* when X=1, namely T=5. Again, each point of the sequence has a time in which we go through it *and* there is a time in which we go through x=1.

3. I think the error is the implicit assumption that we cannot go through an infinite sequence of points. In fact, each one of those points is gone through and we can figure out what time each is passed.

The definition of 'infinite' you use in this case doesn't apply. The sequence of halves that you focus on *does* have a limit: x=1. Every single one of them is smaller than 1, so that is a limit. There are two aspects here:

1. For every point in the sequence, there is a point after that point of the sequence.

2. There is something larger than every point of the sequence.

According to 1, the sequence is unbounded. According to 2, the sequence is bounded. Both are true, but there are two different versions of 'bounded' in use here. There is no contradiction. So which version of 'bounded' do you want to use? is this sequence infinite or not?

I would say it is infinite since there are an infinite sets of point in that sequence. But, for my definition, an infinite set can be bounded and even have an end. The problem is in your ambiguous definition, not in the math.