Galileo's proof that two objects of different weight fall exactly at the same speed is a beautiful example of how the human mind is able to figure out how the world works. Most people assume as a matter of course that heavy bodies fall faster than light ones and indeed they do in some cases. A feather will certainly fall more slowly than a rock. But it was Galileo's genius to see that was only because of air resistance. If you remove that, everything would accelerate equally. As a proof Galileo used the following curious argument :
He imagined a light and a heavy body connected by a string and asked: Does the presence of the light body help or hinder the fall of the heavy body ?

If - as intuition tells us - heavy bodies fall faster than light ones, the presence of the light body would slow down the fall of the heavy body, because it would lag behind and therefore restrain the speed of the heavy object.

However, if we consider the entire package of heavy plus light body together, they are obviously heavier than the heavy body on its own. That means the combination package should fall faster because it is now heavier.

So we arrive at the contradiction that the presence of the light body should both speed up and slow down the fall of the heavy body. That is of course impossible, and the only way we can reconcile it is to suppose that both the heavy object and the light object fall at the same speed !