(See Part 1 for the beginning of this discussion in progress …)

We can compare the sizes of two sets of numbers by finding a one-to-one correspondence between them, but in the case of infinitely large sets, strange things can happen. For example, compare the set of positive integers I = {1, 2, 3, 4, …} with the set of squares S = {1, 4, 9, 16, …}. Every element n in I has a corresponding n² in S, and every n² in S has a corresponding n in I. Here we find that a subset of the set of integers (a subset which has omitted an infinite number of integers) has the same size as the set of all integers.

Playing with the same paradox, Hilbert’s Hotel imagines a hotel that can hold an infinite number of guests. Suppose you ask for a room but the hotel is full. No problem—every guest moves one room higher (room n moves to room n + 1), and room 1 is now free.

But now suppose the hotel is full, and you’ve brought an infinite number of friends. Again, no problem—every guest moves to the room number twice the old room number (room n moves to room 2n), and the infinitely many odd-numbered rooms become free.

Infinity is best seen as a concept, not a number. To understand this, we should realize that zero can also be seen as a concept and not a number. Consider a situation in which I have three liters of water. I give you a third so that I have two liters and you have one. I now have twice what you have. I will always have twice what you have, regardless of the number of liters of water except for zero. If I start with zero liters, I can’t really give you anything, and if I “gave” you a third of my zero liters, I would no longer have twice as much as you.

Not all infinities are the same. Let’s move from integers to real numbers (real numbers are all numbers that we’re familiar with: the integers as well as 3.7, 1/7, π, √2, and so on).

The number of numbers between 0 and 1 is obviously the same as that between 1 and 2. But it gets interesting when we realize that there are the same number of numbers in the range 0–1 as 1–∞.

The proof is quite simple: for every number x in the range 0–1, the value 1/x is in the range 1–∞. (If x = 0.1, 1/x = 10; if x = 0.25, 1/x = 4; and so on) And now we go in the other direction: for every number y in the range 1–∞, 1/y is in the range 0–1. There’s a one-to-one correspondence, so the sets must be of equal sizes. QED.

(Note that this isn’t a trick or fallacy. You might have seen the proof that 1 = 2, but that “proof” only works because it contains an error. Not so in this case.)

The resolution of this paradox is fairly straightforward, but resolving the paradox isn’t the point here. The point is that this isn’t intuitive. Use caution when using infinity-based apologetic arguments.

Let’s conclude by revisiting William Lane Craig’s example from last time.

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.… In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will have already finished.

The problem is that he confuses counting infinitely many negative integers with counting all the negative integers. As we’ve seen, there are the same number of negative integers as just the number of negative squares –1², –2², –3², …. Our mysterious Counting Man could have counted an infinite number of negative integers but still have infinitely many yet to count.

For a more thorough analysis, read the critique from Prof. Wes Morriston.

And isn’t the apologist who casts infinity-based arguments living in a glass house? The atheist might raise the infinite regress problem—Who created God, and who created God’s creator, and who created that creator, and so on? The apologist will sidestep the problem by saying (without evidence) that God has always existed. Okay, if God can have existed forever, why not the universe? And if the forever universe succumbs to the problem that we wouldn’t be able to get to now, how does the forever God avoid it?

This post is not meant as proof that all of Craig’s infinity based arguments are invalid or even that any of them are. I simply want to ask apologists who aren’t mathematicians to appreciate their limits and tread lightly in topics infinite.

Of course, if the apologist’s goal is simply to baffle people and win points by intimidation, then this may be just the approach.

Related posts:

• Part 1 of this topic: Infinity—Nothing to Trifle With
• Word of the Day: Hoare’s Dictum

Related articles:

• “Aleph number,” Wikipedia.
• Wes Morriston, “Must the Past Have a Beginning?” Philo, 1999.
• William Lane Craig, “The Existence of God and the Beginning of the Universe,” Truth Journal.