The kalam cosmological argument rests on the idea that the universe has a beginning; its second premise states as much. Advocates of the argument offer two kinds of argument in favour of this claim: scientific and mathematical. Here three mathematical arguments for the finitude of the past will be outlined.

The first argument draws on the idea that actual infinites cannot exist, the second on the idea that actual infinites cannot be created by successive addition, and the third on the idea that actual infinites cannot be traversed.

If any of these arguments is successful, then the second premise of the kalam arguments will have been proven.

The Impossibility of an Actual Infinite

The first mathematical argument for the claim that the universe has a beginning draws on the idea that the existence of an infinite number of anything leads to logical contradictions. If the universe did not have a beginning, then the past would be infinite, i.e. there would be an infinite number of past times. There cannot, however, be an infinite number of anything, and so the past cannot be infinite, and so the universe must have had a beginning.

Why think that there cannot be an infinite number of anything? There are two types of infinites, potential infinites and actual infinites. Potential infinites are purely conceptual, and clearly both can and do exist. Mathematicians employ the concept of infinity to solve equations. We can imagine things being infinite. Actual infinites, though, arguably, cannot exist. For an actual infinite to exist it is not sufficient that we can imagine an infinite number of things; for an actual infinite to exist there must be an infinite number of things. This, however, leads to certain logical problems.

The most famous problem that arises from the existence of an actual infinite is the Hilbert’s Hotel paradox. Hilbert’s Hotel is a (hypothetical) hotel with an infinite number of rooms, each of which is occupide by a guest. As there are an infinite number of rooms and an infinite number of guests, every room is occupied; the hotel cannot accomodate another guest. However, if a new guest arrives, then it is possible to free up a room for them by moving the guest in room number 1 to room number 2, and the guest in room number 2 to room number 3, and so on. As for every room n there is a room n + 1, every guest can be moved into a different room, thus leaving room number 1 vacant. The new guest, then, can be accommodated after all. This is clearly paradoxical; it is not possible that a hotel both can and cannot accommodate a new guest. Hilbert’s Hotel, therefore, is not possible.

A similar paradox arises if the past is infinite. If there exists an infinite past, then if we were to assign a number to each past moment then every real number (i.e. every postive integer) would be assigned to some moment. There would therefore be no unassigned number to be assigned to the present moment as it passes into the past. However, by reassigning the numbers such that moment number one becomes moment number two, and moment number two becomes moment number three, and so on, we could free up moment number one to be assigned to the present. If the past is infinite, therefore, then there both is and is not a free number to be assigned to the present as it passes into the past.

That such a paradox results from the assumption that the past is infinite, it is claimed, demonstrates that it is not possible that that assumption is correct. The past, it seems, cannot be infinite, because it is not possible that there be an infinite number of past moments. If the past cannot be infinite, then the universe must have a beginning. This is the first mathematical argument for the second premise of the kalam cosmological argument.

The Impossibility of an Actual Infinite created by Successive Addition

The second mathematical argument for the claim that the universe has a beginning draws on the idea that an actual infinite cannot be created by successive addition. If one begins with a number, and repeatedly adds one to it, one will never arrive at infinity. If one has a heap of sand, and repeatedly adds more sand to it, the heap will never become infinitely large. Taking something finite and repeatedly adding finite quantities to it will never make it infinite. Actual infinites cannot be created by successive addition.

The past has been created by successive addition. The past continuously grows as one moment after another passes from the future into the present and then into the past. Every moment that is now past was once in the future, but was added to the past by the passage of time.

If actual infinites cannot be created by successive addition, and the past was created by successive addition, then the past cannot be an actual infinite. The past must be finite, and the universe must therefore have had a beginning. This is the second mathematical argument for the second premise of the kalam cosmological argument.

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.