In Zeno's Paradox, the reader is presented with the clever illusion that you must traverse an infinite number of halves. This is not true. Further, there is the implicit and deceptive suggestion in the paradox that the more halves you must traverse, the longer it will take you to get there. This is also not true. Both of these deceptions are simple category mistakes. We don't traverse halves. Halves are not a measure of distance. We traverse miles. We traverse inches. We traverse meters and centimeters. We don't traverse halves.
Notice that what we actually traverse in Zeno's Paradox above is sixty miles. Sixty miles is a finite amount. Notice further that even when we begin insisting upon going halfway first, and halfway to halfway before that, the number of miles doesn't change at all. It stays sixty miles, a finite amount. In fact, no matter how many times you slice it and dice it, as long as the number of slices is finite, it remains sixty miles.
If you travel at sixty miles per hour when driving from home to your office sixty miles away, you will get there in sixty minutes, or one hour.
If you must go thirty miles first, you will get to the halfway point in thirty minutes, and you will still get to your office in one hour:
30 minutes + 30 minutes = 60 minutes, or 1 hour
If you must go fifteen miles before you can go thirty miles before you can go sixty miles, the equation looks like this:
15 + 15 + 15 + 15 = 60 minutes, or 1 hour
If you must go seven and a half miles before you can go fifteen before you can go thirty before you can go sixty, the equation looks like this:
7.5 + 7.5 + 7.5 + 7.5 + 7.5 + 7.5 + 7.5 + 7.5 = 60 minutes
Notice that no matter how many times you cut each portion in half, you do not change the final distance of sixty miles and you do not increase the amount of time it takes to traverse sixty miles. It always takes sixty minutes.
This is extremely important. As long as the number of halves are not actually infinite, as long as we have not reached "the end of infinity", the distance remains sixty miles and the amount of time remains sixty minutes.
Note also as we approach an infinite number of halves, that the sections of distance decrease. 60 goes to 30, then 15, then 7.5, then 3.75, and smaller and smaller and smaller. As we approach an infinite amount of sections of distance, the size of each section goes toward zero. What this means is that if we could actually reach "the end of infinity", each section would have zero length.
Now, Zeno's claim is that since there are an infinite number of sections, we can never travel sixty miles. The reality is that if there could ever be an actualized infinite number of sections, we would be at the office before we even left our home. Take a look at what the equation would look like if there were a literal infinite number of halves:
0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + ...+ 0 = 0 minutes.
It doesn't matter how many times you add zero to zero... you will never get sixty.
Thus, as long as we agree that it is impossible to get to "the end of infinity", the sixty miles from your home to your office will always be sixty miles. As soon as we claim that it IS possible to get to "the end of infinity", the distance from your home to your office becomes zero miles.
So, we see by analyzing Zeno's Paradox that it is obviously impossible to get to "the end of infinity". Of course, we shouldn't need to analyze Zeno's Paradox to come to this conclusion, because the definition of infinity is "without end". Just look it up in the dictionary.
