According to William Craig’s defense of the kalam cosmological argument, an actual infinite cannot exist. This claim is important not only for Craig’s main claim that the universe had a beginning, but also for a followup response to the suggestion that the universe cannot be part of a wider, infinitely regressive history wherein our universe is only one of infinitely many that have existed. Instead of actual infinites, Craig proposes that only potential infinites can exist. A potential infinite is a collection of things that is finite in size at any given time, but is growing without limit.
So suppose that God has created Zeke and set him to run across an infinite number of flagstones. What God cannot do, according to Craig’s view, is to create Zeke at time t and place him on a path that (at t) already has an infinite number of flagstones on it, since such a path would be actually infinite, and this is what is claimed cannot exist. Instead, what God must do is place Zeke on a path that has some number n of flagstones at t, but then as Zeke runs, more flagstones get added to the path, so that (for example) at time t+1, the path has n+1 flagstones on it, at time t+2 the path has n+2 flagstones on it, and so on.
But what limits God at t, from creating all the flagstones that Zeke is going to run across? Since God can only create a finite number of flagstones at t, suppose Zeke is initially placed at the very end of the path. In order for Zeke to continue his run, God is obliged to create another flagstone at t+1 as Zeke takes his next step. But what prevents an omnipotent God from having already created that flagstone back at t?
If we are to say that God is omnipotent, it seems we should accept the following principle regarding omnipotence and creative power:
If God is omnipotent, then if it is logically possible for God to create x at t, then God can create x at t.
So when God creates Zeke at t and places him at the end of a path consisting of n flagstones, it is logically possible for God to have created, instead, a path consisting of n+1 flagstones at t, or n+2 flagstones at t, or n+3 flagstones at t, and so on.
To put it another way, to hold that God can only create potential infinites, but not actual infinites, is to hold that at any time t, God must create a finite number of things, and that if he wants more things, he can only add them later through successive addition. Although each successive addition to the collection is logically possible for God to add, he cannot add them all at t, but must wait and keep adding them later. But for any one of these additions, if God must wait until after t to create it, then he cannot have created it at t, in which case, by the foregoing principle, God is not omnipotent.
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