Actual Infinity in Reality?

[GrandizerII]

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.

I appreciate all the effort you are putting in, but it seems like you are trying to over complicate things, and answer a number of other question that where not being posed.   It doesn't matter what time the train will arrive in Boston, and the dispute is not that you can reach the end.  

We have a line with a start and an end point.  We will assume that there are an infinite number of points between these two positions.
We progress through this line towards the end point, passing through each point in succession along the way.
From any given point, along that line we will always have more points between the current position and the end position.
All prior points must be reached, in order to reach the end position
If there is always another point, that is not the end, and which precedes the end, then end cannot be reached.
Therefore the end position is not reachable if there is an infinite number of points which must be traveled.  

The disagreement is not that you cannot reach the end position (I believe that motion is fairly well evidenced).  There is not a problem with the logic here.   The problem is that if you have to complete something that never ends, then you will never complete.  To say that it is infinite and that it ends, is contradictory.  Something cannot be both A and !A at the same time.

Ok, perhaps if you let go of your unscientific time "theory", you might see the answer clearly. I think the A-theory of time is your main hurdle right there, whether or not polymath's answer works just as well with the A-theory of time.