Grim Reapers and the Infinite Past

One of the modern arguments for God’s existence is the Kalam Cosmological Argument (see my outline of it here); one of that argument’s two premises is that the universe began to exist. While I covered several reasons to accept that premise in my original post, my favorite reason for it was, unfortunately, too lengthy to include. Having finished the introductory outline of a case for Christianity, I now devote my first “random” post to this reason.

It’s called the Grim Reaper Paradox, and its application to the Kalam is due to the work of philosophers Rob Koons and Alexander Pruss. In order to understand it, though, we’ve first got to talk about the general logical tool known as “proof by contradiction.” The basic idea is that we don’t have to judge ideas as isolated points; we can judge the plausibility of an idea by the plausibility of its implications as well. If an idea has ridiculous implications, we shouldn’t believe it. The proof by contradiction is simply an extreme case of this reasoning; it is absolutely impossible for a contradiction to exist in reality, so if an idea implies a contradiction, that idea is necessarily false. (Fun fact: the simplest way to prove that 2 is irrational is a proof by contradiction.)

Returning to the Kalam Cosmological Argument, our goal is to show that the universe began to exist. Now either it began to exist or it didn’t (it’s “past-eternal”); therefore, one way to show that it began to exist is to assume that it didn’t and derive a contradiction. This is precisely what the Grim Reaper Paradox does. It goes like this:

Assume that our universe is past-eternal. Using conventional dating, we can number every year – this is 2019, there was a year 20, a year 2000 BC, and a year 20000000 BC. Since the universe is past-eternal, there is a BC year associated with every natural number. Now suppose there is also a past-eternal man named Fred. Unfortunately for him, there are infinitely many Grim Reapers out to kill him. There is one Grim Reaper for every natural number, and it is the duty of Grim Reaper number n to kill Fred during the year n BC.

  1. For any natural number n, Fred is either dead or alive at the end of n+1 BC.
  2. If he is dead, he remains dead, and is dead at the end of n BC.
  3. If he is alive, the nth Grim Reaper kills him, and he is dead at the end of n BC.
  4. Either way, Fred will be dead at the end of n BC for any natural number n.
  5. As a special case, we know that Fred is dead at the end of 1 BC.
  6. Because Fred is dead, some Grim Reaper must have killed him; since every Grim Reaper has a number, the one that killed him must have a number. Let’s call it Grim Reaper k, where k is a natural number.
  7. The fact that Grim Reaper k killed Fred means that he was alive after k+1 BC, but since k+1 is a natural number, this contradicts 4.

So we achieved our goal! We assumed that the universe was past-eternal and derived a contradiction; it follows that the universe cannot be past-eternal, i.e. it began to exist. Furthermore, it seems to me that each step of the argument is completely logically unassailable; the only way to avoid it is to say that the contradiction resulted from some other questionable assumption. In the interest of telling a good story, we did make a few such assumptions (e.g. the existence of a past-eternal man and the existence of an infinite number of Grim Reapers). But we can now trim these out to show that the contradiction remains even when the story is more boring.

We can easily replace the infinity of Grim Reapers with a single Grim Reaper who checks every year to see if Fred is dead; the contradiction is then derived in that there is a year in which Fred both was and was not dead. To replace our past-eternal man, there can instead be a past-eternal sheet of paper that each Grim Reaper is to sign during his year if it isn’t already signed. Alternatively, each Grim Reaper can supernaturally add a proton or a planet to the (past-eternal) universe if another Grim Reaper has not already done so. So these peripheral assumptions can be reduced to quite modest ones without damaging the essence of the contradiction; it would seem that the source of the problem really is the assumption of the past-eternal universe. If that’s right, we can be confident that the universe began to exist even apart from the abundance of scientific evidence we have for the same conclusion. In the face of these two lines of evidence, the traditional atheistic view of the universe as eternal no longer holds water. The inescapable beginning of the cosmos is something that any view of reality must explain; I expect that my atheist friends will have a lot more difficulty with that than me.

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