(Cross-posted at Rational Thoughts)

I’m currently working on a nine-part series on the *kalam* cosmological argument and I thought it would be nice to post this particular entry here.

Defenders of the KCA muster several different arguments in support of the premise that the universe began to exist. These arguments are both philosophical and scientific in nature. Arguments under the former category involve showing that the existence of an actually infinite number of things is metaphysically impossible. If the universe never began to exist, then its past duration would be actually infinite. Since actual infinities cannot exist, then the past duration of the universe must have been finite, implying that the universe must have begun to exist. Even if one grants that it is possible for an actual infinite to exist, it still cannot be formed by successive addition, and henceforth the past duration of the universe must still be finite. From a scientific perspective, the beginning of the universe is strongly supported by modern big bang cosmology. The proponent of the KCA thus finds himself comfortably seated in the midst of mainstream cosmology. Combined, these two reasons lend strong support to the truth of the second premise.

**The Impossibility of Actual Infinities**

It’s helpful to distinguish between actual and potential infinities. Potential infinities are sets that are constantly increasing toward infinity as a limit, but never attain infinite status. A more accurate description would be to say that their members are indefinite. An actual infinite, by contrast, is a set x that contains a subset x’ that is equivalent to x. That is, they are *denumerable*. Phrased in layman’s terms, a set is actually infinite if a part of the set is as big as the whole set. A potentially infinite set would thus be a set in which a part is less than the whole. “The crucial difference between an infinite set and an indefinite collection would be that the former is conceived as a determinate whole actually possessing an infinite number of members, while the latter never actually attains infinity, although it increases perpetually. We have, then, three types of collection that we must keep conceptually distinct: finite, infinite, and indefinite. [1] Because it leads to contradictions and absurdities, an actually infinite set cannot exist in reality.

There are several examples which illustrate the absurdity of the existence of an actually infinite number of things, the most famous of which is known as Hilbert’s paradox of the grand hotel. For the sake of clarity, however, I’ll use a simple example used by Craig:

Imagine I had an infinite number of marbles in my possession, and that I wanted to give you some. In fact, suppose I wanted to give you an infinite number of marbles. In that case I would have zero marbles left for myself.

However, another way to do it would be to give you all of the odd numbered marbles. Then I would still have an infinite left over for myself, and you would have an infinite too. You’d have just as many as I would — and, in fact, each of us would have just as many as I originally had before we divided into odd and even! Or another approach would be for me to give you all the marbles numbered four and higher. That way, you would have an infinite of marbles, but I would have only three marbles left.

What these illustrates demonstrate is that the notion of an actual infinite number of things leads to contradictory results. In my first case in which I gave you all the marbles, infinity minus infinity is zero, in the second case in which I gave you all the odd-numbered marbles, infinity minus infinity is infinity; and in the third case in which I gave you all the marbles numbered four and greater, infinity minus infinity is three. In each case, we have subtracted the identical number from the identical number, but we have come up with non-identical results. [2]

The point of this example is that arithmetical operations with actually infinite quantities yield contradictory answers, and thus it is metaphysically impossible for actual infinites to exist. The notion of an actually infinite set is purely conceptual and has no relation to reality. It should be noted here that while one is able to work with actual infinities in set theory and calculus, they existence *in re *is metaphysically impossible. Their existence is only permitted in mathematics because mathematical operations involving infinite quantities are prohibited. In reality, however, there is nothing stopping someone from adding or subtracting from an infinite quantity of marbles.

Suppose however, that actual infinities could exist in reality. Would this serve as a defeater for the second premise? It seems not, for even if actual infinities could exist in reality, they could not be formed by successive addition nor could they be navigated successfully. It is impossible to form an actually infinite quantity by successive addition, as one can always add another number to what they have counted. No matter how many times one adds a number to a finite quantity, one will never yield an infinite quantity.

Even if actual infinities were possible, it is unclear that they could be traversed. Consider Bertrand Russell’s example of Tristram Shandy, who writes his autobiography at such a slow pace that it takes him a whole year to write about a single day. If Shandy had been writing for eternity past, then he would be infinitely far behind. [3] Since it is impossible to traverse an actually infinite past, then we should not have arrived at this point. But since we have, we can conclude that the past duration of the universe was finite.

Critics have sometimes compared the impossibility of forming an actual infinite to Zeno’s paradoxes of motion, which, though tricky and stubborn, are obviously wrongheaded. But these comparisons are not accurate for several reasons: First, the distances traversed in an infinite past are actual and equal, as opposed to being potential and unequal in Zeno’s paradoxes. Second, the distances traversed in Zeno’s paradoxes sum to a finite distance, whereas the distances traversed in an infinite paste sum to an infinite distance. Finally, it begs the question by presupposing the distance traversed as being composed of an infinite amount of points. Critics of Zeno held that the existence of the line itself is prior to any divisions that are made in it. Moreover, in regards to an infinite past, divisions such as a halfway, a quarter of a way, and a third of the way are unintelligible because there is no beginning, unlike in Zeno’s paradoxes. [4]

**Scientific Pointers to a Beginning of the Universe**

** **

Due to the heavy influence of Aristotelianism, scientists and philosophers from the medieval periods up until the early 1900’s firmly believed in the eternality of the universe. The first indications that the universe was not eternal started to surface in 1917 with the advent of Albert Einstein’s general theory of relativity. Einstein himself was a believer in an eternal universe, and when he saw that his theory of general relativity did not permit such a model, he introduced a “fudge factor” into his equations to maintain an eternal universe. By exploiting the shortcomings of Einstein’s model, the Russian mathematician Alexander Friedman and the Belgian astronomer Georges Lemaitre independently developed an expanding model of the universe. Further evidence came in 1929, when astronomer Edwin Hubble confirmed the expanding universe predicted by the Friedman-Lemaitre model through his discovery of redshift. The fact that the universe was expanding implied that in some point in the past, it was compacted together tightly, for if one reverses the expansion of the universe backwards in time, the universe becomes more and more dense until it reaches a state of infinite density. This had the jolting conclusion that the universe, over 14 billion years ago, had once been compressed to a size of an infinitely dense point known as a singularity. Since space and time themselves came into existence at this singularity, it served as a boundary for space-time, as there was no moment “before” the big bang. Hence, the origin posited by the standard big bang model is that of an absolute origin* ex nihilo*.

** **

** **

** **

** **

**
**

**This lent strong support to the Judeo-Christian doctrine of creation ex nihilo. In fact, the bang theory, far from having atheistic implications, was actually criticized for being too religious when first proposed.
**

### Notes:

*The Blackwell Companion to Natural Theology*(Blackwell: 2009) p.105

*The Case for a Creator*(Grand Rapids, MI: Zondervan 2004) p.103

*TBCNT*, pp. 119