In order to understand (2.1), we need to understand the difference between a potential infinite and an actual infinite. Crudely put, a potential infinite is a collection which is increasing toward infinity as a limit, but never gets there. Such a collection is really indefinite, not infinite. The sign of this sort of infinity, which is used in calculus, is ¥. An actual infinite is a collection in which the number of members really is infinite. The collection is not growing toward infinity; it is infinite, it is "complete."
Now (2.11) maintains, not that a potentially infinite number of things cannot exist, but that an actually infinite number of things cannot exist. For if an actually infinite number of things could exist, this would spawn all sorts of absurdities.
What absurdities is Craig referring to?
Let us imagine a hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. When a new guest arrives asking for a room, the proprietor apologizes, "Sorry, all the rooms are full." But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are full. There is not a single vacant room throughout the entire infinite hotel. Now suppose a new guest shows up, asking for a room. "But of course!" says the proprietor, and he immediately shifts the person in room #1 into room #2, the person in room #2 into room #3, the person in room #3 into room #4 and so on, out to infinity. As a result of these room changes, room #1 now becomes vacant and the new guest gratefully checks in. But remember, before he arrived, all the rooms were full! Equally curious, according to the mathematicians, there are now no more persons in the hotel than there were before: the number is just infinite. But how can this be? The proprietor just added the new guest's name to the register and gave him his keys-how can there not be one more person in the hotel than before? But the situation becomes even stranger. For suppose an infinity of new guests show up the desk, asking for a room. "Of course, of course!" says the proprietor, and he proceeds to shift the person in room #1 into room #2, the person in room #2 into room #4, the person in room #3 into room #6, and so on out to infinity, always putting each former occupant into the room number twice his own. As a result, all the odd numbered rooms become vacant, and the infinity of new guests is easily accommodated. And yet, before they came, all the rooms were full! And again, strangely enough, the number of guests in the hotel is the same after the infinity of new guests check in as before, even though there were as many new guests as old guests. In fact, the proprietor could repeat this process infinitely many timesand yet there would never be one single person more in the hotel than before. (Taken from http://www.leaderu.com/truth/3truth11.html)
Craig's overall point is that because an absolute infinity results in absurdities then it is metaphysically impossible. Another argument against the existence of actual infinity is the problem of traversing infinity. Let's say an infinite amount of time as existed. Time could never arrive at this present moment because an infinite amount of time would have to elapse first. But since infinite means no beginning and no end, by definition, it cannot be elapsed.
Now, with that established, let's proceed to the actual argument. Let's take the apostle Peter and use him as our example. According to Christianity, when we die as believers, we will go to heaven and that is where we will spend eternity. So taking Peter as our example, after he died for the faith he went to heaven to be in the presence of God and that is where he will reside at forever. When we say that Peter will reside in heaven forever or for eternity, what are we referring to? According to Craig, it means that Peter will be in heaven for a potential infinite amount of time, not actual. But is this really the case?
Craig elsewhere has given the example of a man jumping from one stone to another. Each time that the man jumps a new stone is created for him to land on. According to Craig, the process can continue forever and it will always remain a potential infinite, because there will always be a new stone created for the man to hop to. Looking at our example of Peter again, Craig is certainly correct. All the days that Peter resides in heaven will always be a potential infinite because there will always be "the next day", just as how there will always be another stone in our above example.
But when it comes to God's omniscient perspective of Peter's after life, we run into a problem. Craig would certainly agree that God knows all things, which would include all of Peter's days in heaven. But if an actual infinite amount of days is impossible then God's knowledge of all of Peter's afterlife is limited. In Craig's mind, Peter's days in heaven is limited since there will always be "that next day". So if God does not have a knowledge of a actual infinite amount of days for Peter's life then that means that there will always be "that next day" that God does not know about. Let's look at this slightly different to help drive the point home. Let's represent Peter's days in his afterlife with dots. Each dot represents a single day in his afterlife and as the next day comes a new dot is created. What is God's knowledge on these dots? Does God see a potential infinite amount of dots that keep getting longer and longer? If so then there will always be future dots that God does not know about since what God sees is only dots coming into existence or a line of dots that continues to grow, not an actual infinite amount of dots.
A possible theistic objection to this argument is that it is a false dichotomy. And that it is not a matter of either God's knowledge comprising an potential infinite amount of days vs. an actual infinite amount. The objector could then proceed to state that God's knowledge of Peter's after life days is finite, not necessarily because there are future days that God does not know about, but because Peter's after life itself is finite. This way of looking at it says that God does not know of a finite amount of dots that keeps growing with no knowledge of anything beyond those finite dots that are already in existence. Rather, God knows of every dot that comes into existence.
This argument is implying that we cannot put a set value on God's knowledge of Peter's days in heaven. But neither will the argument affirm that the amount of days is actual infinite. It is somewhat paradoxical. If we cannot ask the question of how many days of Peters after life that God knows about then this means that God's knowledge in its entirety is not set.
The problem with this objection is that there is always a set value with anything finite. A potential infinite amount of days is a finite amount of days that has a beginning and has an end. It cannot be said that God knows of every after life day that comes without a limit because being without a limit is actual infinite, not a potential infinite. The objector would be inadvertently affirming that God knowledge of all the days of Peter's after life is an actual infinite which leads us to our final point:
Someone could reject Craig's argument against actual infinity and say that God does see an actual infinite complete set of dots, not a finite amount that keeps growing. If in God's knowledge there exists an actual infinite amount of days of Peter's life then one might ask: How is it that God knows of something that Peter himself will never experience? On Peter's level, he will never experience an actual infinite amount of days in heaven, it will always be potential infinite. If God knows of an actual infinite amount of days then his knowledge does not reflect reality. God would know something that would never occur, hence, his knowledge regarding Peter's afterlife would be inaccurate, and he would not truly be omniscient. Either way we cut it, true omniscience is out of the question.
In light of the above, I conclude that an omniscient God cannot exist.
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