Consider a ball falling toward earth. why is the momentum of the ball not conserved? in what system that includes the falling ball is the momentum conserved? "

First, lets correct your concept. Momentum is still conserved in the system. the ball is falling towards th earth and the earth is also accelerating towards the ball. the forces betwwen the earth and the ball are the same by Newton's third law. but the rate of acceleration of the earth is neglectable seemingly to have no momentum, beacuse the earth has higher mass than the ball by F=ma. Therefore momentum is still conserved.