(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 *n*^{2} in **S**, and every *n*^{2} 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 2*n*), 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 –1^{2}, –2^{2}, –3^{2}, …. 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.

To Bob S,

Whatever may happen in the abstract realm of mathematics is one thing. But the real world is another thing. The concept of the infinite does not literally apply to the real world.

I disagree with Craig’s argument though. I do think he has a wrong understanding of the meaning of the infinite. However, a better argument can be built.

And of course that the universe may be eternal. But in that case, it would be cyclical down to the minutest details.

“The concept of the infinite does not literally apply to the real world.”

How can you make such a bold claim wish such confidence?

“And of course that the universe may be eternal.”

doesn’t this contradict your previous statement that the infinite does not apply to the real world?

“But in that case, it would be cyclical down to the minutest details.”

This seems much less believable to me than an infinite series of universes. Or perhaps a single universe that eventually experiences heat death and just persists forever in a very boring manner. Or, quite possibly, something else that I have never heard of.

How can you so confidently state that a cyclical universe is the only possibility?

If the universe were infinite, what would it mean? There would be an infinite amount of planets like Earth with infinite replicas of Bobs the atheists saying the same thing or else things slightly different, or maybe planets in which Bob is a believer or planets in which Bob was never born, planets in which the human species does not exist, yet chimpanzees do, and so on. Is this believable?

But a logical problem is that if there is an infinite amount of particles in the universe, and this is implied by the infinity of the universe, is that amount even or odd? It makes no sense.

Besides, if there is an infinite amount of particles, some of them must be at a finite distance from here, but some others at an infinite distance. But then, where are the boundaries between the two groups?

So the universe must be finite.

But if the universe is finite, yet eternal, and that it has no goal, how can be anything but cyclical? Do you have a better model?

You speak of a heat death. Fine, but if the universe is eternal, it should have come to that heat death by now, and for an infinite time.

“But a logical problem is that if there is an infinite amount of particles in the universe, and this is implied by the infinity of the universe, is that amount even or odd? It makes no sense.”

Again, the question your asking makes no sense. “infinity” is not an integer, therefore, asking whether it is even or odd makes no sense. “Is the color blue even or odd?” makes exactly as much sense as “Is infinity even or odd?”

“if there is an infinite amount of particles, some of them must be at a finite distance from here, but some others at an infinite distance.”

Wrong. Every pair of particles is some finite distance apart from one another. If there is an infinite number of particles and an infinite amount of space, it is possible that the particles are arranged in such a way that no ball of finite radius can encompass all of the particles. However, each particle is at some position in space, and any pair of them is some distance apart. Saying 2 points are infinitely far apart doesn’t really make sense. (I suppose it might make some kind of sense if you are in a disconnected manifold or something, but for simplicities sake I’m assuming we are in regular 3 dimensional space)

I’m not really sure what you mean when you say “finite yet eternal”. If you are saying that there are only finitely many states of matter and there is an infinite amount of time, then perhaps you are right that a cyclical universe becomes necessary. I don’t think we have evidence of either of those premises though.

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Bob,

Thanks for these posts. I love infinities and always enjoy reading about them. I also agree with your assessment of what is wrong with WLC’s argument.

It seems to me that the question that WLC is really asking here is “if a man has been counting infinitely into the past, if he stops the first time he reaches infinity when will he stop?” This of course makes no sense. The nature of infinity does not allow for “the first time” types of statements usually.

To put another way, he is saying that this guy has counted an infinite number of things. If I chop off a finite number of them (that is really what he is going when he asks why the guy didn’t finish yesterday) then you still have an infinite number of things. Well yeah, that is how infinity works.

Hausdorff:

Glad you liked them! I jotted down the rough idea of my outrage about WLC’s casual forays into infinity arguments years ago, but trying to round this out into a coherent pair of blog posts was hard. This pushes me into new areas. I’m sure someone who was seriously qualified could add much to this conversation. But then this gets back to my main point: that this stuff is counterintuitive and that laymen should tread lightly.

«Wrong. Every pair of particles is some finite distance apart from one another.»

If by “every” you mean an infinite amount, then the problem is merely moved a step further, but in no way settled. But if by “every” you mean “a finite amount”, then you must grant that the universe is finite.

Again, don’t lose sight of the difference between the mathematical realm, in which many weird entities exist, and the physical realm, in which there are only natural numbers.

If there is a finite amount of particles in the universe, then yes, there must be a finite amount of possible states, which means that in the eternity, each of these states has taken place an infinite amount of times. However, to be more specific, in a cyclical universe, there is not a real addition of cycles. What happens is that time goes forward a bit, then bounces back and goes backward to its starting point, undoing everything it had done, then bounces back and resumes the cycle. But of course, it is only metaphorically that I use the word “then” in this description.

“«Wrong. Every pair of particles is some finite distance apart from one another.»

If by “every” you mean an infinite amount, then the problem is merely moved a step further, but in no way settled”

By every pair, I mean, given any 2 particles, those 2 particles are a finite distance from one another. Let me try to explain in a different way, I think we are just talking past each other.

First, suppose we are in a situation where there is an infinite amount of space. Let’s assume for simplicity that we are in a simple 3 dimensional space, we can pick any point in space to be the origin, from there every other point in space can be described by three coordinates in 3 mutually perpendicular directions. We usually describe this as (x,y,z).

Since we are assuming that there is an infinite amount of space, x,y, and z can take on any value. Since we have an infinite amount of space, we can fit an infinite number of particles into that space. Every point lies on this grid somewhere. So suppose we are some at some point on this grid, every other point is somewhere else on the grid, and therefore every other point is some finite distance from us. No other single point is an infinite distance away. If we are at the origin, and we want to figure out how far any point is from us, we just have to look at the coordinate, say (a,b,c) and we know that this particular point is sqrt(a^2+b^2+c^2) from us.

When you say some particles are an infinite distance away from each other, I don’t know what you mean. I don’t think it makes sense. How can one point be an infinite distance away from another?

“Again, don’t lose sight of the difference between the mathematical realm, in which many weird entities exist, and the physical realm, in which there are only natural numbers.”

I do understand that there is a difference between mathematical theory and the real world. I think it is also an overstatement to say it is impossible for infinities to exist in the real world in every situation. And when you say there are only natural numbers in the real world, you are wrong. Natural numbers don’t even include fractions, are you saying fractions can’t exist in the real world? Have you never cut something in half? I’m not sure you are expressing what you think you are when you say there are only natural numbers, can you elaborate?

Well, you may be right, I need to think about it.

But there is still the other objection: is the amount of particles even or odd? I don’t think the question is absurd or irrelevant. It seems pretty clear to me that any quantity I am faced with in the physical world MUST be either even or odd. For instance, if a believer says that God has an infinite amount of thoughts, then he, who can count, MUST know whether they are even or odd. But of course it makes no sense, so God cannot have literally an infinite amount of thoughts.

In the realm of mathematics, things may be different though. In fact, a great deal of progress was made in maths when people realized that mathematical entities had other laws than physical entities. It took long to even hint at the concept of zero.

I am not convinced by Craig’s arguments though. I think he uses the weakest of the arguments of this category.

To Hausdorff,

(Are you German?)

I thought about it carefully, and here is my reply.

If we assumed that particles were mathematical points in your infinite space, then yes, you would be right, we would not need “infinite distances”.

The problem is that each particle takes up some space, so that only a finite amount can fit into a finite space with finite distances between them. If there is an infinite amount of them, then at least one dimension must stretch into infinity, which is what I mean by “infinite distance”. We may imagine any finite distance we want in the three dimensions, it would still be the case that only a finite amount of particles can fit there. Precisely because they are not mathematical points but physical things.

I see what you are saying, if we have an infinite number of particles, each of which takes up at least some volume, then we need an infinite amount of space, at least in one direction. I agree with that.

As to the even or odd thing, let’s look at the definition of even and odd:

an integer n is even if there is some integer k such that n=2k

an integer m is odd if there is some integer L such that m=2L+1

infinity isn’t an integer so it doesn’t really fit into this.

An no, I’m not German. We just happened to be studying hausdorff spaces in topology class the day I made my main character in WoW and it just wound up being the name I use everywhere online

To Hausdorff,

I gather that you have a strong mathematical background.

This is just what we need in such a debate.