Irrational numbers cannot exist

Let's look at the number line. You can see -5,-4,-3,-2,-1,0,1,2,3,4 and so on. If you look between 0 and 1 there is going to be a rational number 1/2 exactly in the middle. There is going to be a rational number 1/4 exactly in the middle of 0 and 1/2. You can do this for any two numbers and make the distance between them smaller and smaller.

So you can fill up the whole number line by cutting it in halves. Therefore, the whole number line is made up of rational numbers and irrational numbers cannot exist.

>>6
You didn't prove that there are no holes.  You merely proved there are an infinite number of rational numbers.  However, infinity is not enough to fill up even the number line between 0 and 1.  Cantor proved that there are an uncountable (for all intents and purposes, "uncountable" is a larger quantity than "infinity") number of real numbers.  He also proved that there are a countable (but infinite) number of rational numbers.  The gap between infinite and uncountable is filled in by transcendental numbers (irrational numbers that are non-algebraic).