Aristotle's argument for the existence of the unmoved mover progresses as follows:
1) There exists movement in the world. 2) Things that move were set into motion by something else. 3) If everything that moves were caused to move by something else, there would be an infinite chain of causes. This can't happen. 4) Thus, there must have been something that caused the first movement. 5) From 3, this first cause cannot itself have been moved. 6) From 4, there must be an unmoved mover.
