About Two Objections to the Kalām Argument

How does a beginningless universe entail an actual infinite?

William Lane Craig is well-known for establishing the Kalām Argument within the God’s Existence debate. That argument is so well-known in Academia and to the public, and so criticized as praised by many people. But why? It seems that the premises of the argument seem obvious and, in the same way, not obvious. Normally, the argument’s structure goes like this:

1. Whatever begins to exist has a cause.

2. The universe begins to exist.

3. Therefore, the universe has a cause.¹

At first sight, as the first premise is evident, I will not deal with it now (probably in another case). The problem is with the second premise. How do to demonstrate that the universe has a beginning? Well, appealing to the impossibility of a universe with an infinite history. Certainly, the Christian faith tells us that God has created the world, and it is by faith that I know that the universe began to exist and, therefore, has a cause, namely, God. But how do I know that the universe began to exist by reason? Saint Thomas says that “Novitas mundi habetur tantum per revelationem. Et ideo non potest probari demonstrative.”² It can be obstinate to defy the authority of St. Thomas, but we can give our bona fides to the contemporary philosophers to try to prove it. Having said this, Craig gives two philosophical arguments (in addition to scientific evidence, but I will pass off it). The first ─and this argument is that I will be interested in here─ is The Impossibility of an Actuality Infinite Number of Things:

1) An actually infinite number of things cannot exist.

2) A beginningless series of events in time entails an actually infinite number of things.

3) Therefore, a beginningless series of events in time cannot exist.³

In the development of the premises, Craig makes a distinction between two types of infinity: potential infinite and actual infinite. The former is like a set increasing towards the infinite as if the infinite was the limit, but it never gets to that limit; in reality, this sort of infinity is potentially long; it can add things without limit. “Such a collection is really indefinite, not infinite”⁴, Craig says. The actual infinite, on the other hand, is actually long, not potentially. It’s, in fact, already, with all the members of its set. That infinity is problematic if it exists in reality, according to Craig. To demonstrate it, he takes the example of Hilbert’s Hotel:

As a warm-up, let’s first imagine a hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are occupied. When a new guest arrives asking for a room, the proprietor apologizes, “Sorry, all the rooms are full,” and that’s the end of the story. But now let us imagine a hotel with an infinite number of rooms and suppose once more that all the rooms are occupied. 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 already occupied! But the situation becomes even stranger. For suppose an infinity of new guests shows up at the desk, each asking for a room. “Of course, of course!” says the proprietor, and he proceeds to shift the person in room #l 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 with a number twice his own. Because any natural number multiplied by two always equals an even number, all the guests wind up in even-numbered rooms. 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 already occupied! In fact, the proprietor could repeat this process infinitely many times and so always accommodate new guests, despite the fact that the hotel is completely full!⁵

Because of the strange consequences of Hilbert’s Hotel, Craig concludes that the hotel is absurd, and therefore “It logically follows that if such a hotel is impossible, then so is the real existence of an actual infinite.”⁶ And that’s the actual infinite: the existence here and now of an infinite number of things, and for that reason, it is impossible in reality. We can grant the first premise. The question now is about the second. How does a beginningless series of events in time be an actually infinite number of things? Craig takes for granted that it’s obvious since “if the universe never began to exist, then before the present event there have existed an actually infinite number of previous events.”⁷ The second premise of this philosophical argument is now central, we will examine whether it is correct that an infinitely old universe entails an actual infinite through two objections to it.

An Infinitely Old Universe ≠ Actual Infinite?

Edward Feser⁸ mentions that examples of situations impossibles (metaphysically speaking), like Hilbert’s Hotel, don’t show us that applies to the universe with an infinite past or history. In fact, he notices that amount of infinity of hours, minutes, seconds, or days and years has not any relation to an actually infinite number of things. Why? Whereas the actually infinite number of things exist together now, and not after or before, they are now, past moments of the universe don’t exist anymore. If that were the case (if the past moments exist), then it would be equal to the actual infinite. This is further accentuated by the presentism, Craig holds this position. But the problem isn’t presentism, rather, is how does an infinite series of years, days, etc. (or an infinite series of events in time of the universe) count as an actual infinite?

What we are discussing is a beginningless universe, a universe with an infinite past. Such a universe goes towards the infinite, just like it has been since infinity. Therefore, it is a potential infinite, and it’s impossible to future becomes an actual infinite. But whether we regress far back in time, can the same be said in the past? Apparently, yes. Because while we regress far back in time, we cannot get to the limit; if it was an actual infinite, all the members of the set (all the past moments) would exist at the same time, and there isn’t an exponential increase.

It can be that with actually we refer that the actual infinite series are uncountable. For example, an actual infinite set of natural numbers: we say that it extends meanwhile we count; in this sense, we say that it’s an actual infinite. Clearly, in that sense, the mathematician speaks of the actual infinities; not about concrete objects, but numbers and abstract objects. Now, that amount of infinite events in the past, even though the amount isn’t numerable, isn’t an actual infinite (at least in the principal sense that we discuss here). It is the same with mathematicians’ actual infinities. So, it isn’t appropriate to speak of this sort of set in the principal sense, because their members aren’t concrete objects, and, no matter how many members have, it will never be an actual infinite set as long as it’s just about abstract objects.

So, in order to function Craig’s argument, he has to demonstrate the relationship between an infinitely old universe and an actual infinite, it doesn’t work with abstract objects, but with concrete (existent) objects.

Traversing the Infinity?

Someone could say that a proof of the incoherence of an infinitely old universe, it’s the impossibility of going through infinity.

Of course, if we start to count all natural numbers, a collection with infinite numbers, we cannot cover all the members. Now, we can extrapolate to the universe. Given an infinity of past moments or events, the universe ─in order to arrive at the present─ had to traverse that number of infinite moments, but the infinity is not possible to traverse (it has been demonstrated with the example of natural numbers). In addition to this, that set of events appears to be an actual infinite (that’s what we discussed before). Therefore, a universe with an infinite history and the actual infinite series cannot exist.

The Catholic apologist Jimmy Akin,⁹ for his part, makes three distinctions of infinity with the natural numbers:

The set of positive natural numbers has a limit at the beginning (i.e., the number 1) but no final limit. It just keeps going {1, 2, 3 . . .}.

The set of negative natural numbers as a limit at the end (i.e., the number -1), but no starting limit, so it goes {. . . -3, -2, -1}.

The set of all natural numbers (which also includes 0) has neither a beginning limit nor an ending limit, so it goes {. . . -3, -2, -1, 0, 1, 2, 3 . . .}.

In that argument above, which he calls “Traversing an Infinite”, exists an error or fallacy: that series isn’t infinite, but it passes as one at first sight. We will suppose that the universe began to exist, but with an infinite past and with a present. The opponent could argue with the same line, with the same argument, and he could think that the argument function very well. However, a universe with a beginning, infinitely far back in time is not, after all, infinite. It doesn’t fulfill the second sort of infinity. Nor the first, because it has a finish, namely, the present moment. Not even the third! In this case, due to that assumption, the argument commits the First-and-Last Fallacy. This fallacy is defined like this: “The First-and-Last Fallacy occurs if and only if a person envisions a supposedly infinite series as having both a first and a last element.”

Naturally, someone could say, “but that happens with a beginningless universe?” The fallacy tells us that a set with a first and last element is not infinite. So that a beginningless universe doesn’t have a first element (or moment in this case), but it has a last element, the present moment. This universe coincides with the second sort of infinite, so it’s really infinity. Also, this class of universe is the subject discussion of Craig. In fact, the philosophical argument’s second premise speaks about “a beginningless series of events in time”, and he tries to prove that it is impossible. We will now examine it.

Akin’s objection, so that it works, has to prove that series is not infinite. But that isn’t possible. He assumed that an infinite set with a last element, but without having a first element, is infinite. So, it seems that the fallacy doesn’t apply to it. In contrast, the Traversing an Infinite argument seems to work. Due to the infinity of the set, it’s possible to cover all the members (the events) of it, as much as we can cover a lot of them, but never cover all the infinite members. However, let’s leave to count, let’s talk about the possibility of the arrival at the present in that sort of universe.

Clearly, a series of events in time is an addition from one event to another. One moment in time is the actual present at that moment, with a duration, more or less small. The present is a moment in time, just like a past moment or a future moment. Now then, the A-Theory of time tells us that the events in time aren’t all equally real, the future and past don’t exist, only the present and the things that exist in it. So, this theory leads us to consider time as a succession of events at successive intervals of time. The present arrives when prior past events at successive intervals of time have passed. For example, a is a present event-state, b is a past event-state, and c is a future event-state. Let J a collection of events in time: […, b, b’, a, c, c’, …]. So that a arrives, have to pass b and b’, one after one. That’s been a succession of intervals in time.

Now then, since we are considering a beginningless universe, it’s possible for a succession of intervals in time to the present in such a universe? It seems not. Akin himself recognizes that a series, so that be traversed, needs two extremes: the start and the final (an obviousness). In fact, he cites Aquinas,¹⁰ who says:

For, although the infinite does not exist actually and all at once, it can exist successively. For, so considered, any infinite is finite. Therefore, being finite, any single one of the preceding solar revolutions could be completed; but if, on the assumption of the world’s eternity, all of them are thought of as existing simultaneously, then there would be no question of a first one, am, therefore, of a passing through them, for, unless there we two extremes, no transition is possible.

In an infinite set, with infinite events, so that be traversed and the time can arrive in a present event, it needs to pass over one after one event. But they are infinite! And never comes to the present, and it is the final of the set, to which the time never arrives. Moreover, the distance of time between one event and the next is infinitely large. This improves the second premise of the philosophical argument because this argument we have discussed shows that a beginningless universe (with an infinite history and infinite events in time) is equal that an actual infinite. Therefore, a universe without a beginning is not possible, due to the impossibility of an actual infinite existing in reality.

So, the Traversing an Infinity argument seems to be sound.

An actually actual infinite?

Despite the conclusion above, however, it seems to me that the question is still unclosed. Why? First, let’s check something that Craig says. In the same place where he treats the alleged likeness of the argument against the possibility of traversing the infinity and Zeno’s paradoxes, he says: “Whereas in Zeno’s thought experiments the intervals traversed are potential and unequal, in the case of an infinite past the intervals are actual and equal.”¹¹ Apparently, within an infinite past the intervals of time are real (and the events within them) in the same way, in the same degree. Hence Feser’s claim on the alleged actual of that series, in which he removes this actuality.

But, in another sense is “actual.” Let’s consider again the Impossibility of Traversing an Infinity argument. In that argument, it’s proposed the travel of time from an interval of time to the present. Obviously, that’s impossible, given the impossibility of traversing infinity. However, those members of the set (infinite past) are actually actual? To check that, I will take two perspectives, “at the present” (or “perspective from the last element”) and “from the infinity.” If we take the perspective “at the present”, the past events do not exist anymore (if we were presentist), by this perspective Feser critiques the philosophical argument’s second premise. But if we take the perspective “from the infinity”, the infinite number of events “make an effort” to reach the present, in this sense the infinite events are “actual”; that perspective seems to be used by Craig in his argumentation.

This series makes this “effort” in order to be formed, but it will never get to infinity. In fact, the second philosophical argument that is presented by Craig is The Impossibility of Forming an Actually Infinite Collection of Things by Adding One Member after Another. A series of events in time, within existing a succession of intervals of time and, so, an addition of one event after another, is not formed. An infinite set with infinite members that will never be formed totally (in its infinite extension) isn’t an actual infinite. Hence Craig’s conclusion of dismissing the set of events in time as actually infinite.

He reaches the same conclusion as Feser: a series of events in time, as much as they are infinite, doesn’t equal an actual infinite.¹²

In these two ways, it’s proven that a series of infinite events in time is not an actual infinite. Now, the Traversing an Infinity argument seems to be ineffectual in demonstrating that a series of infinite events in time is an actual infinite. (At the most, it demonstrates that it’s an actual infinite in the mathematician’s sense or a sort of potential infinite). Even though it’s impossible to traverse a such set of events in time by an infinite number of events and the infinite duration of them, now have appeared two different reasons for demonstrating that a series of infinite events in time is not an actual infinite: I) the past events don’t exit, unlike the present (in an actual infinite all the members already exist); II) a true actual infinite is not be formed, because it is already formed.

Therefore, given the not entailment of a beginningless universe to an actual infinite, the first philosophical argument is not conclusive. So, at the moment, the second premise of the Kalām Argument has not been proven. Anyway, there are other arguments that intend to prove the second premise, like the Grim Reaper Paradox. But, for the moment, that’s all.

1. Craig, W. L. (2008). Reasonable Faith: Christian Faith and Apologetics (3.a ed.). Crossway Books. p.112

2. “The newness of the world is known only by revelation; and therefore it cannot be proved demonstratively.’’ (S.Th., I, q.46, a.2, sed contra.)

3. Craig, 2008, p.116.

4. Craig, 2008, p.116.

5. Craig, 2008, p.188.

6. Craig, 2008, p.119.

7. Craig, 2008, p.120.

8. Feser, E. (September 2, 2016). A difficulty for Craig’s kalām cosmological argument? Edward Feser. https://edwardfeser.blogspot.com/2016/09/a-difficulty-for-craigs-kalam_2.html; and Feser, E. (September 8, 2016). Yeah, but is it actually actually infinite? Edward Feser. https://edwardfeser.blogspot.com/2016/09/yeah-but-is-it-actually-actually.html

9. Akin, J. (November 12, 2020). The Kalaam Cosmological Argument and the First-and-Last Fallacy. Jimmy Akin. http://jimmyakin.com/2020/11/the-kalaam-cosmological-argument-and-the-first-and-last-fallacy.html. For now, my interest resides in the first argument that he discusses in his post. Also, he addresses the Grim Reaper Paradox and the Eternal Society Paradox.

10. Contra gentiles, II, Ch.38, 11.

11. Craig, 2008, p.122.

12. The difference is that Craig uses this conclusion (of the second argument) to prove that the universe without a beginning is impossible, because ─according to him─ the series of events in time needs to be an actual infinite in order to exist in a beginningless universe.