The Cosmological Argument (part 2)

August 20, 2007     Time: 00:28:01

We are going to continue our discussion of the cosmological argument. We left off by talking about premise (1) which is “whatever begins to exist has a cause.” You will remember that I said there are both metaphysical as well as scientific reasons to believe in the truth of this premise. Metaphysically it seems to me to be simply an obvious first principle of metaphysics – being does not come from non-being. Therefore any attempt to prove the principle would be based upon premises or assumptions that would be less obvious than the principle itself. I suggested that one could give various illustrations of this in order to bring this across to your unbelieving friend that you are sharing the argument with.

But if one did want to argue for this metaphysically, what one could point out for example is that if things could really come into being without causes then there is no explanation why just everything and anything doesn’t come into being out of nothing. It seems more plausible to believe that things that come into being are caused to do so. We saw then as well that this is empirically confirmed constantly in science.

We did consider one objection to the premise based upon modern quantum physics. According to some interpretations of quantum physics there are events on the quantum level that are spontaneous events that do not have deterministic causes. But I suggested that this is not a bona fide counterexample to this first premise because that is only the case on some interpretations of quantum physics. On others even subatomic reality is fully deterministic and therefore the detractor of this first premise is assuming without justification the truth of an indeterministic interpretation of quantum physics.

I also suggested that in any case, even on the indeterministic interpretations, things don’t really come into being out of nothing. Rather, these particles that originate on the subatomic level spontaneously are fluctuations of the vacuum energy that exists on that level. Therefore they don’t really represent things coming into being from nothing, and the same holds of theories of the universe which posit the universe’s origin out of the quantum vacuum. In none of these cases, properly speaking, are we really talking about things coming into being uncaused out of nothing. So these are not bona fide counterexamples to the truth of the first premise.

Other atheists that I’ve read in the literature have said that the first premise (whatever begins to exist has a cause) is true only for things in the universe but it is not true of the universe itself. Anything that comes into being in the universe has a cause but it is not true that if the universe itself comes into being it has to have a cause. I think that this objection misconstrues the nature of this first premise. This first premise does not state merely a physical law like, say, the law of gravitation or the laws of thermodynamics which are valid only for things in the universe. Rather, premise (1) is not a physical principle. Premise (1) is a metaphysical principle. That is to say, it is a principle that is true of being as being; namely, being cannot come from non-being. Something cannot come out of nothing uncaused. Therefore this principle would apply to all of reality. You cannot dismiss this principle like a taxi cab when you get to the origin of the universe itself. If the universe came into existence then I think it is plausible that it would have to have a cause.

Even the late J. L. Mackie, who was one of the most prominent atheists of our time, admitted that he found the idea that the universe popped into being uncaused out of nothing incredible. He commented, “I myself find it hard to accept the notion of selfcreation from nothing, even given unrestricted chance. And how can this be given, if there really is nothing?”[1] Just think about Mackie’s point.[2] On the atheistic view, prior to the Big Bang there isn’t even the potentiality of the universe existing because there just isn’t anything prior to the Big Bang. So there isn’t even the potentiality of the universe existing prior to the Big Bang. Nothing is prior to the Big Bang. But then how could the universe become actual if there wasn’t even the potentiality of its existence? It seems to make much more sense, I think, to say that the potentiality of the universe lay in the creative power of God to bring the universe into being.

So, on balance then, it seems to me that the objections to this first premise are not at all plausible and the premise itself is more plausibly true than its contradictory. Therefore I think that premise (1) is well established.

That brings us to the second premise – “the universe began to exist.” This is really the crucial premise of the argument. As you will see, I offer four arguments in favor of the beginning of the universe. The first two are purely philosophical arguments; the second two are scientific confirmations of the conclusion reached by the philosophical arguments.

Let’s begin by looking at the first philosophical argument that the universe began to exist. It goes like this:

1. An actually infinite number of things cannot exist.

2. A beginningless series of past events involves an actually infinite number of things.

3. Therefore, a beginningless series of past events cannot exist.

Let me define some crucial terms in order to make this argument clear. When we speak about an actually infinite number of things, this needs to be distinguished from what mathematicians and philosophers call a potential infinite. So first and foremost we need to distinguish between a potential infinite and an actual infinite. A potential infinite is a collection which is at every point finite but always growing toward infinity as a limit. So this collection is really an indefinite collection. It is finite at every point in time you pick but it grows ceaselessly forever and therefore it grows toward infinity as a limit. It never arrives at infinity but it approaches infinity endlessly. So this is why it is called potentially infinite. The infinite serves merely as a limit toward which the series grows, but it never arrives there. The symbol of this sort of infinity is the so-called lazy eight. This is the symbol of a potential infinite – ∞.

By contrast, an actual infinite is a collection which has an actually infinite number of members in it. That is to say, the number of members in the collection exceeds any natural number that you can think of. The natural numbers are the numbers 0, 1, 2, 3, and so forth on out to infinity. So to say that a collection has an actually infinite number of members means that the number of its members is greater than any natural number that you could arbitrarily pick. So it is not growing toward infinity; it is infinite. There exists an actually infinite number of things in the collection. The symbol that mathematicians use for this type of infinity is the Hebrew letter aleph which looks like this – א.

So when we say in the first philosophical argument that an actually infinite number of things cannot exist, we are speaking here of the impossibility of the existence of an actual infinite. But we are not denying the existence of a potential infinite. Potential infinites can exist. For example, the distance between any two points such as on this podium could be divided in half, and then in half again, and then in half again, on and on to infinity. But you would never arrive at infinity. The podium would never be actually divided into an infinite number of bits. Infinity would merely serve as the limit toward which you could endlessly divide.[3] So we don’t want to deny the existence of potential infinities. The question here is whether an actually infinite number of things can exist.

I think that the easiest way to show that an actually infinite number of things cannot exist is simply to give some illustrations of the absurdities that would result if an actually infinite number of things could exist in reality. For example, if you could have an actually infinite number of things in reality then you could subtract various quantities from your infinite collection. That will lead to self-contradictory results. Let me give an example. Suppose we have the numbers 1, 2, 3, out to infinity. Suppose we subtract from this collection all of the odd numbers. What we will have left then would be all of the even numbers: 2, 4, 6, etc. out to infinity. So in this case we have subtracted an infinite number of odd numbers from an infinite number of natural numbers and we got an infinite number of even numbers. So infinity minus infinity is infinity. But let’s suppose that we had instead subtracted from the series of natural numbers all of the numbers greater than 4. So we subtract 4, 5, 6, out to infinity. In this case, if we subtract from the serious of natural numbers all numbers greater than 3 you have only three natural numbers left, namely, 1, 2, and 3. So you subtracted an infinite number of numbers from an infinite number of numbers and you get 3! So infinity minus infinity is 3! You could get any answer you want from 0 to infinity and yet in each case you will have subtracted identical quantities from identical quantities. This leads to simply self-contradictions.

Therefore, in mathematics, subtraction of infinite quantities is prohibited because it leads to self-contradictions. But obviously if an infinite number of things existed in reality, you couldn’t prevent somebody from taking some of them away. For example, if I had an infinite number of marbles I could give some to you. I could give you all of the odd numbered marbles, or I could give you a few handfuls. In that case you are going to have these kinds of self-contradictions result.

Let me give one of my favorite illustrations of the absurdities that would result from the existence of an actually infinite number of things. This is called Hilbert's Hotel after the great German mathematician David Hilbert. Hilbert first invites us to imagine a hotel with a finite number of rooms. He says let's suppose that all of the rooms are full, and suppose that a new guest shows up at the desk asking for a room. “Sorry,” the manager says, “all of the rooms are full.” And the new guest has to be turned away. But, Hilbert says, now let's suppose instead we have an infinite hotel with an infinite number of rooms. Let's suppose again that all the rooms are full. This is critical to understand. There is not a single vacant room in the entire infinite hotel – every room is occupied by some guest. Now let's suppose that a new guest shows up at the desk asking for a room. “Of course! Of course!” says the manager. And he proceeds to shift the guest who was in room 1 into room 2, the guest who was in room 2 into room 3, the guest who was in room 3 into room 4, and so on and so forth out to infinity. As a result of these transpositions, room 1 now becomes vacant and the new guest gratefully checks in. And yet, before he arrived, all the rooms were full. Even stranger, according to the mathematicians, there are no more people in the hotel than there were before the new guest checked in. But how can this be? We just saw the manager give him his keys and walk down the hall. How could there not be one more person in the hotel than before? But Hilbert's Hotel becomes even stranger. Now let's suppose, Hilbert says, that an infinite number of new guests arrive at the desk asking to check in. And remember – all the rooms are full. Every room is occupied.[4] “No problem! No problem!” says the manager. And he moves the guest who was in room 1 into room 2, the guest who was in room 2 into room 4, the guest who was in room 3 into room 6, moving every former guest into the room number twice his own. Since any number multiplied by 2 gets you an even number, all of the odd numbered rooms become vacant! And the infinity of new guests gratefully check in. Yet, before they arrived all the rooms were full. Again, according to the mathematicians, there are no more people in the hotel than before they checked in, even though there were just as many new guests as there were old guests. In fact, a proprietor could do this an infinite number of times and there would always be room for more guests and there would never be any more people in the hotel than before.

But Hilbert's Hotel is even stranger than the great German mathematician gave it out to be. For let's apply these operations of subtraction now to this hotel and ask ourselves what happens when some of the guests check out? Let's suppose that all of the guests in the odd numbered rooms check out. Well, in that case an infinite number of people have left the hotel and yet, according to the mathematicians, there is still just as many people in the hotel as there were before they checked out. And if the proprietor doesn't like the hotel looking half empty – every odd numbered room vacant – he can reverse the process before and lo' and behold the hotel will be full again, crammed to the gills. You might think then you could just keep subtracting people infinitely from the hotel and there would never be any less persons. Ah! But that wouldn't be correct! For if all of the people in rooms 4, 5, 6, [up to infinity] checked out of the hotel, at once the hotel would be virtually empty. The infinite would be reduced to finitude and the guest register would have only three names left on it. And yet it would be the case that the same number of guests checked out this time as when all of the odd numbered guests checked out.

Now, Hilbert's Hotel is absurd. Does anybody think that such a hotel could exist in reality? As one student said to me, if Hilbert's Hotel could exist it would have to have a sign outside that said, “No Vacancy. Guests welcome!”

So I think these kinds of illustrations illustrate that the existence of an actually infinite number of things leads to evident absurdities. Infinity is simply an idea in your mind. It is not something that exists in reality. Once you understand the concept of infinity you can think up these stories yourself that can be very clever about the paradoxes and absurdities that result from the existence of an actually infinite number of things. Sometimes, for example, I like to talk about a library that has an infinite number of books in it and what would happen if you checked out every other book. Or if you checked out all the books greater than a certain number, and so on and so forth. Or you could think about marbles or baseball cards or something. Any of these sorts of things can be used to illustrate the kinds of absurdities that result if an actually infinite number of things could exist.

What objections are raised to this first premise since obviously not everybody agrees with it? The typical objection that you will hear raised against this premise is that modern mathematical set theory proves that an actually infinite number of things can exist. For example, in set theory the set of all the natural numbers (0, 1, 2, 3, etc.) is said to be an infinite set. It contains an actually infinite number of members in the set. So obviously an actually infinite number of things can exist; namely, there are an actually infinite number of numbers. This is the very typical refutation that is given to this first premise. But I think that this first objection is all too easy. It is too quick.

1. Not all mathematicians agree that actual infinites exist even in the mathematical realm. Some mathematicians would regard series like 0, 1, 2, 3, ... as merely potentially infinite. They simply grown toward infinity as a limit but they never actually get there.[5] That is a minority view in mathematics – I want to be quite honest about that. But I don't think anybody has ever proven that there are an actually infinite number of numbers as opposed to saying the number series is potentially infinite. So this is a respected minority position in mathematics, and therefore it is far to quick and easy to just say, “Oh, well, there are an infinite number of numbers and so this first premise is false.”

2. Secondly, and more importantly, existence in the mathematical realm does not imply existence in the real world. When mathematicians say that there is an infinite set of numbers, say, they are simply postulating a realm of discourse which is governed by certain arbitrarily adopted axioms and rules which are just presupposed. Then you can talk consistently about such collections given these axioms and rules. But there is no guarantee at all that the axioms are true, or that the rules are true, or that an actually infinite number of things can exist in the real world. It may well be the case that in set theory, given its axioms and its rules, you can talk consistently about infinite quantities, but that is no guarantee and no proof at all that these entities actually really exist.

3. In any case, the real existence of an actually infinite number of things would violate the rules of set theory. As I've said, subtraction with infinite quantities leads to self contradictions. Therefore, infinite set theory simply prohibits these kinds of operations in order to preserve consistency. You simply are not allowed to subtract or divide with infinite quantities because it leads to self-contradictions. So you can slap the mathematician's hand if he tries to divide or subtract infinite quantities, but if an actually infinite number of things could exist in the real world then there is nothing that would prevent us from breaking this arbitrary rule. If I had an actually infinite number of marbles or baseball cards, I could subtract from them or divide them as I please. And that will lead to these sorts of absurdities. So I think this just illustrates again that the rules which govern this mathematical realm of numbers and sets and so forth don't have any necessary applicability to the real world in which we live.

Therefore, this is not a good refutation or counterexample to the premise (1) – that an actually infinite number of things cannot exist.

Other times people will dispute this premise by saying we can find examples of an actually infinite number of things in the real world. For example, they will say, isn't every finite distance between two points capable of being divided into one-half, one-fourth, one-eighth, and so on out to infinity? Doesn't that prove that within any finite distance there is an infinite number of sub-intervals or parts? Well, I think that this is, again, a fallacious objection that confuses a potential infinite with an actual infinite. It is true that you can continue to divide any finite distance as long as you want. You can keep going on and on and on, dividing into smaller and smaller parts. But that doesn't prove that you have an actually infinite number of parts already there. That just shows that the number of divisions is potentially infinite. Infinity serves as the limit which you will endlessly approach in your divisions but you will never actually arrive there. Now, if you are assuming already that any finite distance is composed of an actually infinite number of parts then you are begging the question. You are assuming what the example is supposed to prove, namely, that any finite distance is composed of an actually infinite number of parts. You cannot just presuppose that. So it seems to me that as long as we think of any distance as being simply potentially infinitely divisible then there is no difficulty posed by saying that you can divide forever and ever and ever.[6] That doesn't prove that it is actually composed of an actually infinite number of parts.

Those are the principal objections that are brought against this premise (1). Therefore, I don't find either of those to be powerful or persuasive objections. It seems to me that in view of the absurdities to which an actually infinite number of things would lead we are justify in believing that an actually infinite number of things cannot exist.

That leads to the second premise – “a beginningless series of past events involves an actually infinite number of things.” I think this premise is uncontroversial. If there were a beginningless series of past events (the universe never began to exist, the series of past events just went back and back forever) then there would exist an actually infinite number of past events. Therefore you would have an actually infinite number of things existing in reality; namely, an actually infinite number of past events. So this second premise, I think, is uncontroversial. If the universe never began to exist but has existed from infinity in the past then there will have been prior to today an actually infinite number of past events.

But if an actually infinite number of things cannot exist, and a beginningless series of past events involves an actually infinite number of things, then the conclusion follows:

3. Therefore a beginningless series of past events cannot exist.

In other words, the series of past events must have a beginning. It cannot be beginningless. There must be a finite number of past events prior to today. Therefore the series of past events must have had a beginning.

Since the universe is not distinct from the series of past events – the series of past events is just past states of the universe – it therefore follows that if the series of past events began to exist, the universe must have begun to exist. And that is the second premise that we set out to prove.

Therefore, I find this first philosophical argument to be a good argument for the beginning of the universe.[7]



[1] J. L. Mackie, Times Literary Supplement [5 February, 1982], p. 126.

[2] 5:05

[3] 10:03

[4] 15:00

[5] 20:01

[6] 25:00

[7] Total Running Time: 27:45 (Copyright © 2007 William Lane Craig)