Paradoxes

The first paradox that I remember hearing about was in high school physics class.  I'm recalling from memory that it went something like this: In a race with a tortoise and the tortoise is given a head start, the runner can never catch up to the tortoise.  Why?  Consider this: at each point in the race, the runner will make up half the distance between himself and the tortoise.  Then afterwards, he will again make up half of the remaining distance.  Subsequently, as the race continues -he at a certain point between himself and the tortoise -will continue making up half of the remaining distance between himself and the tortoise.  But since half of the remaining distance is equal to a certain distance which is always greater than zero, he can never catch up because 1/2 of something always yields something remaining.

A solution to this paradox was invented.   (I think)  It became known as the concept of a mathematical limit. For example, as the denominator of a fraction gets bigger, the number gets smaller.  As the denominator tends toward infinity, the fraction tends toward zero.  Voila!  The concept of a mathematical limit is born, and the paradox is solved.  With the concept of a limit, the runner will catch the tortoise because distance remaining will tend toward zero.

I don't know if there is a moral to this story.  Maybe if someone gives you an impossible riddle as paradox, you can solve it if you invent something.  But your adversary who gave you the riddle in order to vex you may not accept your invention.  Then what?  A punch in the nose, perhaps?

What brought this thought on?  There seem to be paradoxes all around.  Solutions to problems seem to exist, yet the problems remain.  Is this not a paradox?  That is, if a solution seems to exist, but the problem remains, is this not a paradox?  What accounts for the lack of a solution?

Update: a few minutes later

Now these poll results mentioned in the National Review article by Jim Geraghty seem like a paradox.  How is the Tea Party responsible for what the madman does or what the Democrats say?  Yet that is my take on the poll.  Those participants in the poll are holding the Tea Party responsible for these things.  How could this poll be interpreted any other way?  Yet, I would reject that interpretation.  That would appear to be a paradox.