One of the two philosophical arguments which is supposed to show that the history of the universe must be finite is the impossibility of forming an actual infinite by successive addition. I think this argument begs the question, because there is one premise which can only be true if we assume that the conclusion is true.
- The argument
The argument, which can be seen here, looks like this:
1. A collection formed by successive addition cannot be actually infinite.
2. The temporal series of past events is a collection formed by successive addition.
3. Therefore, the temporal series of past events cannot be actually infinite.
I am going to grant premise 2 for the sake of the argument, although I think it could be questioned. All that I want to focus on is premise 1. This premise, it seems to me, can only be regarded as true if we assume that the conclusion is true.
First, what is ‘successive addition’? It means nothing more than continually adding one over and over again, 1 + 1 + 1 …, which is itself akin to counting whole numbers one at a time, 1, 2, 3 … . The idea is that such a process can never lead to anything but a finite result, as Craig explains:
“…since any finite quantity plus another finite quantity is always a finite quantity, we shall never arrive at infinity even if we keep on adding forever. Infinity in this case serves merely as a limit which we never attain.”
2. The counterexample
There is obviously a close connection between numbers and our concept of time. Exactly what that relationship is, doesn’t matter too much here. One thing that seems obvious though is that we routinely associate sequences of whole numbers with durations of time. Consider the convention which says that this year is 2019. What this means is that if there had been someone slowly counting off integers one per year since year 0, by now he would have counted up to the number 2019.
Adapting this familiar idea, we can postulate that there is some metronomic person counting off whole integers one every minute. After three minutes he will have counted up to the number 3, after one hundred minutes he will have counted up to the number 100, etc.
Let’s make this very simple and intuitive idea slightly more formal. Let us think of a counting function for this person. It takes an input, x, and returns an output, y. The value of x will be some amount of time that has passed (three minutes, one hundred minutes, etc), and the value of y will be whatever number has been counted to (3, 100, etc).
This counting function is therefore akin to asking the question:
‘After x units of time have passed, which number have they counted to?’
The value of y will be the answer to the question.
When Craig says “any finite quantity plus another finite quantity is always a finite quantity”, we can cash this out in our function as saying something like:
If the value of x is finite, then the value of y is finite.
No matter how much time has passed, so long as it is a finite amount of time, then the number that has been counted to must be merely some finite number.
However, what happens if the value of x is not finite (i.e. if it is transfinite)? Let’s suppose that the amount of time that has passed is greater than any finite amount, i.e. that an infinite amount of time has passed. Will the value of y still remain finite? Clearly, the answer here is no. After all the finite ordinal numbers comes the first transfinite ordinal number, ω. If the amount of time that has passed is greater than any finite amount, but less than any other transfinite amount, then the number that will have been counted to will be ω. That is, the function we have been using so far returns this value if we set the value of x to the right amount. But what this seems to say is that if you had been counting for an infinite amount of time, then the number you would have counted to would be greater than any finite number.
At no point have we specified that we are not using successive addition, i.e. counting. WE have explicitly said that this is what we are doing. All we have varied is how long we have been doing it for. The lesson seems to be that if you only count for a finite amount of time, then you cannot construct an actual infinite by successive addition, but if you do it for an actually infinite amount of time, then you can.
Thus, in order for premise 1 to be considered correct, we have to restrict the amount of time we spend counting to arbitrarily high finite amounts of time. If we place that restriction on, then the premise looks true. But if we take this restriction off, then the premise is false, as we just saw.
This means that whether the premise is true or false depends on whether we think that the value of x can be more than any finite number or not. And that just means whether the extent of past time can be infinite or not. If it can be, then we have enough time to have counted beyond any finite number. Yet, that is the very question we are supposed to be settling here. The conclusion of the argument is that the past is finite. Yet we need to suppose precisely this proposition in order to make the first premise true. Without it, the first premise is false.
Thus, the argument seems to simply beg the question here.