Actual Infinity in Reality?

[polymath257]

The contradiction is to the assumption that there is a last number before 1. That was a proof that there is not.

This is called a proof by contradiction. You assume something, get a contradiction, and thereby show it is false. In this case, the assumption was that there is a last number before 1. There isn't. But there is no reason to think there would be.

If you are saying that you never reach the destination (A), then I would agree.  It was my understanding that you where saying that A was reached, as opposed to Zeno's conclusion that motion was impossible or you have your contradiction.

Here is my reasons for there needing to be an last point (note reasons, not an assumption). 

So we take a line between two points (0 and our destination) 
We'll define A as any point >= the destination, making any point less than the destination !A

We start at point 0 (!A)  and progress through the line of points (either finite or infinite) where the current point is always greater than the previous point.
1.If you reach the destination (A) then the last point you passed where !A is your final point.

I believe that your equation where you progress through where X<Y<A  is analogous to Zeno's paradox.   Where Y is the average between X and A;  and you keep repeating this sequence where then Y becomes X and you solve for Y again.

This involves the first principles of the Law of identity, the law of non-contradiction, and the law of excluded middle applied as we move along our line.  It assumes that we are progressing along the line, and that we do reach the destination.   We start off !A and progress through the point along the line until we reach A.

Now what myself and Steve have been saying, and that a number of people seem to not want to do the math on, is that given this approach; you will never reach A.  And I would agree, given that you never reach A, there will not be a last point prior to A (it's a nonsensical question in that case).  The contradiction would be in saying that both A and !A are true.  And what makes your claim of infinity also prevents you from ever ending at point A.

I think that we can take for granted, we are presented this in the middle.  Can I place another number in there... Sure.   However the problem arises when you start doing so on the ends, and realize that you either cannot finish or cannot start your journey with such thinking, and lead Zeno to conclude that motion is impossible.  Zeno according to the tale was a strict logician and even though it was demonstrated that motion was in fact possible, demanded that he be shown logically where he was wrong.

I hope that we are in agreement, that motion is possible. To me and others, Zeno's problem is not in that math, nor in his logic.  His problem was with his starting assumption of a continuum or having to cross an infinite.

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I think we reach the point A, but not by stopping at each point in that sequence. To stop would entail spending more than an instant at each point, and that would not allow us to reach the end, 1.

In that process, at no stage is the final point reached. But that isn't the complete description of how we get from 0 to 1. There is more there than just that process (which would finish in a finite amount of time).

In 1, you are assuming there is a last point. The proof above shows there is not. yes, we start at !A and proceed to A. But there is not a last point before A. Where is the contradiction?

There is no A and !A being true. We do reach that last point by going through each point in the sequence AND ALSO 1. If we limit ourselves to that sequence, we do not get to A, but we are not limited to being on that sequence. We do go through every point of the sequence *and* 1 also.