This is a paste from the Philosophy of Relgion website (www.philosophyofreligion.info).
Man, I love philosophy – I can’t wait to get this bachelor’s done, so I can get my Masters, then I can get my Doctorate in Philosophy… probably won’t happen till after I’m dead though….
The Impossibility of an Actual Infinite that has been Traversed
The third mathematical argument for the claim that the universe has a beginning draws on the idea that actual infinites cannot be traversed.
If I were to set out on a journey to an infinitely distant point in space, it would not just take me a long time to get there; rather, I would never get there. No matter how long I had been walking for, a part of the journey would still remain. I would never arrive at my destination. Infinite space cannot be traversed.
Similarly, if I were to start counting to infinity, it would not just take me a long time to get there; rather, I would never get there. No matter how long I had been counting for, I would still only have counted to a finite number. It is impossible to traverse the infinite set of numbers between zero and infinity. This also applies to the past. If the past were infinite, then it would not just take a long time to the present to arrive; rather, the present would never arrive. No matter how much time had passed, we would still be working through the infinite past. It is impossible to traverse an infinite period of time.
Clearly, though, the present has arrived, the past has been traversed. The past, therefore, cannot be infinite, but must rather be finite. The universe has a beginning. This is the third mathematical argument for the second premise of the kalam cosmological argument.