Is This a Paradox?

[RArvay]

Everyone here is surely aware of the mathematical assertion that there are an infinite number of finite integers. The proof of this is that for every integer n, there is n+1 yielding the new “n.” This recursion can be repeated endlessly, and thus the proof.

However, there seems something intuitively wrong with the concept that there can be an infinite number of finite integers. Perhaps the seeming paradox lies in definitions.

The fallacy I see in the recursive algorithm (n yields n+1 yielding the next n) is that n+1 is always a finite integer. It is always recurved a finite number of times, never reaching an infinite number.

In another forum I questioned this, and someone well versed in mathematics pointed out my error. He said that I was thinking of infinity as a mathematical value that can be reached. He corrected me by saying that you never get to infinity—not by counting in finite increments.

To me, that is just the point. Since the recursive algorithm cannot get to infinity, it never demonstrates an infinite number of integers. Even though it demonstrates an endless sequence of them, the very endlessness of it prevents the demonstration.

Herein lies the problem with definitions. Infinity is equated with endlessness, a sensible enough definition at first. My error, which I think is not really an error, is to think of infinity as a value that can be reached—not by counting in finite increments, but all at once.

For example, any finite line segment has an infinite number of geometric points along its length. The value of infinity is already present in the number of points. The instant one draws any finite line segment, he has already reached an infinite number of points.

True, one cannot count to it. A point, having dimensions of zero, cannot be incremented to any integral value. You cannot start at the beginning point, label it as one, and then reach point number 2 (or n+1), because you will not have moved past the beginning point.

What one can do, however, is leap to the final endpoint, which would be infinity if one could sequentially number the points. That final value, infinity, is not a finite number, and therefore does not represent a finite integer.

In the real universe, this mathematical problem is of no account if the smallest possible increment of space turns out to be finite, and not a pure geometric point.

In pure mathematics, however, it seems that an endless recursion is not the same as infinity, since the recursion never reaches infinity.

[minkwe]

What is a "finite integer"? What is a point?