These are not exactly paradoxes. They are just trick word problems which play on mathematical concepts, and only appear as paradoxes.
For example... The Racetrack Paradox is just a play on division and infinitesimals. He asks you to divide the race track in half, and run to that
point. Then divide the remaining half in half, and run to that point. Then divide the remaining quarter in half, and run to that point, etc. He is
just asking you to divide the remainder in half, over and over again.
Each time you divide the remainder in half, you end up with a smaller remainder. Sooner or later the remainder becomes so infinitesimally small that
you don't even appear to be moving, but you are technically still moving closer to the finish line. But, you will never reach the finish line while
doing this because of one simple fact...
Mathematically (on paper) you can divide something infinite times. So you will always have an remainder, even if its infinitesimally small. In terms
of the racetrack, you will always stop short of the finish line, infinitely. But that is only on paper (and in the mind) as a concept. In reality,
when the remainder of the racetrack left to the finish line is 0.999999999999999999... you would just round up, and consider the race complete because
its such an infinitesimally small value that you can't tell the difference.
This concept of the racetrack is often used to "prove" 0.999... repeating is equal to 1. Which I don't agree with, because it is not, because you
always have an infinitely small remainder left. But math nuts will talk about "
limits"
and other concepts which are just wrong. Our decimal system lacks the ability to represent 0.999... and most people lack the ability to understand
infinite.
You can read more about it here:
en.wikipedia.org...
-------
When you step back and take a good look at the racetrack "paradox", you see why it's not a paradox. You are just being asked to do a repetitive
task (infinitely divide remainder), and are being told this repetitive task is equal to running a lap around a racetrack (which is not true), and it
forms the illusion of a paradox.
The problem with it is, you can't equate running a lap around a racetrack with running and stopping an infinite amount of times.