This argument comes in three varieties: the Thomistic cosmological argument, the Leibniz cosmological argument, and the Kalam cosmological argument. It is the Kalam argument that has, by far, received the most attention in the literature. So, let’s talk about the Kalam argument.
I give you this piece by William Lane Craig to review.
http://www.leaderu.com/truth/3truth11.html
In it he defends the following argument for the existence of God.
1. Whatever begins to exist has a cause of its existence.
2. The universe began to exist.
3. Therefore, the universe has a cause of its existence.
For Craig, the crucial part of this argument is the second premise. He provides two arguments in support of (2). The first argument is based on the impossibility of an actual infinite and goes as follows.
2.11 An actual infinite cannot exist.
2.12 An infinite temporal regress events is an actual infinite.
2.13 Therefore, an infinite temporal regress of events cannot exist.
If the universe had no beginning, then there would be an infinite regress -- an infinite number of events leading up to our present time. Craig thinks this is impossible. To illustrate this, Craig has us think about Hilbert’s Hotel, the famous thought experiment from mathematician David Hilbert.
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.
And this can be repeated an infinite number of times. You could have an infinite number of new guests arrive without increasing the total number of guests in the hotel. Likewise, you could have an infinite number of guests leave without decreasing the total number of guests in the hotel. Craig thinks some of these consequences are absurd and therefore, we cannot have an actual infinite.
These results are absurd because they are a violation of what he calls Euclid’s maxim. A whole is greater than any of its parts. But when Craig talks of Euclid’s maxim, what he really wants is the following principle:
* A set must have a greater number of elements than any of its proper subsets.
But why should we assume that this principle is true? Well, it cannot be true because it gives rise to these absurd results for it is exactly the acceptance of this principle that makes these results “absurd.” This dilemma is resolved by showing that there are such sets that do (or could) have an infinite number of members.
Take a chunk of space for example. Space can have an infinite number for sub-regions. Even though it is true that we cannot actually subdivide space into an infinite number of parts, there is no problem in thinking of space as having sub-regions prior to any possible divisions.
Craig also fails by assuming that basic algebra works the same when applied to the number of elements in a set. Craig tries to do this and ends up with “absurdities” as we see from the following.
But Hilbert's Hotel is even stranger than the German mathematician gave it out to be. For suppose some of the guests start to check out. Suppose the guest in room #1 departs. Is there not now one less person in the hotel? Not according to the mathematicians-but just ask the woman who makes the beds! Suppose the guests in room numbers 1, 3, 5, . . . check out. In this case an infinite number of people have left the hotel, but according to the mathematicians there are no less people in the hotel-but don't talk to that laundry woman! In fact, we could have every other guest check out of the hotel and repeat this process infinitely many times, and yet there would never be any less people in the hotel. But suppose instead the persons in room number 4, 5, 6, . . . checked out. At a single stroke the hotel would be virtually emptied, the guest register reduced to three names, and the infinite converted to finitude. And yet it would remain true that the same number of guests checked out this time as when the guests in room numbers 1, 3, 5, . . . checked out. Can anyone sincerely believe that such a hotel could exist in reality? These sorts of absurdities illustrate the impossibility of the existence of an actually infinite number of things.
Let’s see what’s going on here. Let m = the number of total rooms in the infinite hotel, let n = the number of odd numbered rooms in the hotel, and let p = the number of rooms 4 or higher. Craig’s reasoning is as such:
(m − n) = infinity, whereas (m − p) = 3.
But, n = p (since both n and p are infinite).
And since n = p, we are subtracting the same number from m. But infinity ≠ 3. Therefore, we have a contradiction.
But as Wes Morriston shows, operations within set theory don’t work this way.
http://stripe.colorado.edu/~morristo/craig-on-the-actual-infinite.pdf
In fact, when we take the difference of two sets, we are not subtracting numbers at all. In logical terms, the difference of two sets A and B is the following.
A − B = {x: x ∈ A and x ∉ B}
In words, A − B is the set of all the elements in A that are not in B. Let’s try this in our case.
m − n = {1, 2, 3, 4,....} − {1, 3, 5, 7,...} = {2, 4 ,6, 8,...}
m − p = {1, 2, 3, 4,....} − {4, 5, 6, 7,...} = {1, 2, 3}
If we then talk about the number of elements a set, we are dealing with the cardinality of the set.
The cardinality of m − n is aleph-null. (Alpha-null is the smallest infinite cardinal number, and by definition, the cardinality of the set of natural numbers.)
The cardinality of m − p is 3
No logical inconsistency.
Craig seems to want to subtract the “numbers” n and p each from m and then show the absurd results. You cannot do this. The cardinalities of m, n, and p are each alpha-null and alpha-null minus alpha-null is left undefined in set theory.
Let’s move on to Craig’s second argument in support of (2). Craig wants to show that even if an actual infinity is possible, the series of past events cannot be of this sort. His argument is as such.
2.21 A collection formed by successive addition cannot be actually infinite.
2.22 The temporal series of past event is a collection formed by successive addition.
2.23 Therefore, the temporal series past events cannot be actually infinite.
Craig’s reasoning is that an infinite collection formed by successive addition could never be completed. It is the problem of what is called “traversing the infinite.”
One cannot form an actually infinite collection of things by successively adding one member after another. Since one can always add one more before arriving at infinity, it is impossible to reach actual infinity.
The first premise of this second argument (2.21) is the one worth challenging. It appears to be false.
Consider Zeno’s paradox. Zeno was the ancient philosopher who had an interesting argument for why motion is impossible. His argument is as follows. In order to go from point A to point B, one needs to first reach the halfway mark and then the halfway mark of the remaining half then the halfway mark of the final fourth, then of the final eighth, and so on ad infinitum. Zeno concluded that therefore, motion is impossible because in order to reach any destination, one would first have to traverse and infinite number of finite distances which would be impossible. But this is exactly where Zeno got it wrong. It is possible to traverse an infinite number of finite distances in a finite amount of time! We cannot blame Zeno for this mistake because in his time, the concept of infinity was not well understood. But of course motion is possible. And today, we can easily solve this problem with the knowledge of how to handle an infinite series.
But Craig insists that we cannot have a beginningless series of events that ends in the present.
...suppose we meet a man who claims to have been counting from eternity and is now finishing: . . ., -3, -2, -1, 0. We could ask, why did he not finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished by then. Thus, at no point in the infinite past could we ever find the man finishing his countdown, for by that point he should already be done!
There is a problem with Craig’s reasoning. He is confusing the task of counting infinitely many numbers with the task of counting all the negative numbers up to zero. The man would have completed the first task but not the second. It is this first task of counting infinitely many numbers that is relevant and this should be sufficient. Why should this task of counting all the negative numbers up to zero be necessary? There could still be a beginningless series of events that ends in the present.
I have given some arguments against premise (2), on philosophical grounds, that the universe began to exist. There is also the argument for (2) on scientific grounds.This is a big topic in itself and I would like to leave this for later so we can spend more time on it. We already have a lot going on right here.
We skipped over the first premise (1) that whatever begins to exist has a cause of its existence. Craig thinks that this self-evident and needs little argument in support of it. But surprisingly, there might be reasons to doubt this. In quantum mechanics we have this idea of vacuum fluctuations where particles pop into existence uncaused. We can discuss this more when we get to the topic of scientific cosmology.
http://universe-review.ca/R03-01-quantumflu.htm
Wes Morriston gives a good critique of Craig’s argument. I recommend you read it.
http://stripe.colorado.edu/~morristo/kalam-not.html
I tried so far to give a summary of his main points.
OK, so what if Craig’s argument is sound? Let’s assume the truth of both (1) and (2) for argument’s sake. This gives us the conclusion Craig argues for, namely, that the universe has a cause. But does this mean that there is a creator? The philosopher, Quentin Smith, presents the case in which it does not. (This is not found online. His argument is presented in a piece titled “Kalam Cosmological Arguments for Atheism” from The Cambridge Companion to Atheism. If anyone else wants to read this, email me for the PDF.)
According to science, the universe began with a big bang 15 billion years ago. But also, according to the science, there cannot be a first time t=0. This is because at this first instant of time, the universe would be in an impossible state and would have to be described by some nonsensical mathematical statements.
What we can discuss is the first interval of time in which the universe began to exist. And since there is no time t=0, this interval is open at the beginning (there can be a boundary point at the end but not at the beginning). Time is continuous. In other words, in this interval of time we have an infinite number on time instances. Think of an instance as a point in time along this time interval (an instance has no temporal duration). Between any two instances there are an infinite many other instances.
So let x be an instance within the first second of the universe existing. Then x is greater than 0 and less than or equal to 1.
If we think about the universe in this way, we can say that every instance of time is preceded and caused by earlier instantaneous states. In this sense, the universe can be self caused. But what if we ask about the cause of the interval as a whole? Well, says Smith, it doesn’t make sense to ask such a thing. The interval is a set which is an abstract object and cannot have causal relations with other objects.
I understand that this is some heavy stuff. Realize that set theory is an essential part of these arguments. And also realize that these arguments are essential in trying to understand the philosophy dealing with the origin of the universe.
