Infinity—Nothing to Trifle With

Infinity—Nothing to Trifle With August 5, 2014

I’m preparing a presentation for the Atheist Alliance of America conference in Seattle, August 7–10. I’ll be posting less often for a week. Thanks for your patience.

The topic of infinity comes up occasionally in apologetics arguments, but this is a lot more involved than most people think. After exploring the subject, apologists may want to be more cautious.

William Lane Craig scatters tacks on the road

Philosopher and apologist William Lane Craig walks where most laymen fear to tread. Like an experienced actor, he has no difficulty imagining himself in all sorts of stretch roles—as a physicist, as a biologist, or as a mathematician.

Since God couldn’t have created the universe if it has been here forever, Craig argues that an infinitely old universe is impossible. He imagines such a universe and argues that it would take an infinite amount of time to get to now. This gulf of infinitely many moments of time would be impossible to cross, so the idea must be impossible.

But why not arrive at time t = now? We must be somewhere on the timeline, and now is as good a place as any. The imaginary infinite timeline isn’t divided into “Points in time we can get to” and “Points we can’t.” And if going from a beginning in time infinitely far in the past and arriving at now is a problem, then imagine a beginningless timeline. Physicist Vic Stenger, for one, makes the distinction between a universe that began infinitely far in the past and a universe without a beginning

Hoare’s Dictum is relevant here. Infinity-based arguments are successful because they’re complicated and confusing, not because they’re accurate.

One of Craig’s conundrums is this:

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.

More on infinity

Before we study this ill-advised descent into mathematics, let’s first explore the concept of infinity.

Everyone knows that the number of integers {1, 2, 3, …} is infinite. It’s easy to see that if one proposed that the set of integers was finite, with a largest integer n, the number n + 1 would be even larger. This understanding of infinity is an old observation, and Aristotle and other ancients noted it.

But there’s more to the topic than that. I remember being startled in an introductory calculus class at a shape sometimes called Gabriel’s Horn (take the two-dimensional curve 1/x from 1 to ∞ and rotate it around the x-axis to make an infinitely long wine glass). This shape has finite volume but infinite surface area. In other words, you could fill such a container with paint, but you could never paint it.

A two-dimensional equivalent is the familiar Koch snowflake. (Start with an equilateral triangle. For every side, erase the middle third and replace it with an outward-facing V with sides the same length as the erased segment. Repeat forever.) At every iteration (see the first few in the drawing above), each line segment becomes 1/3 bigger. Repeat forever, and the perimeter becomes infinitely long. Surprisingly, the area doesn’t become infinite because the entire growing shape could be bounded by a fixed circle. In the 2D equivalent of the Gabriel’s Horn paradox, you could fill in a Koch snowflake with a pencil, but all the pencils in the world couldn’t trace its outline.

Far older than these are any of Zeno’s paradoxes. In one of these, fleet-footed Achilles gives a tortoise a 100-meter head start in a foot race. Achilles is ten times faster, but by the time he reaches the 100-meter mark, the tortoise has gone 10 meters. This isn’t a problem, and he crosses that next 10 meters. But wait a minute—the tortoise has moved again. Every time Achilles crosses the next distance segment, the tortoise has moved ahead. He must cross an infinite series of distances. Will he ever pass the tortoise?

The distance is the infinite sum 100 + 10 + 1 + 1/10 + …. This sum is a little more than 111 meters, which means that Achilles will pass the tortoise and win the race.

Some infinite sums are finite (1 + 1/2 + 1/4 + 1/8 + … = 2).

And some are infinite (1 + 1/2 + 1/3 + 1/4 + … = ∞).

(And this post is getting a bit long. It will be concluded in part 2.)

To say that that having a degree in theology
makes you qualified to say that God is real,
is like claiming that having memorized all the Harry Potter books
makes you qualified to say that unicorns exist.
— Unknown

Photo credit: Wikipedia

(This is an update of a post that originally appeared 7/12/12.)

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  • MNb

    That’s what religion can do to you – it has robbed WLC of all the fun ….
    I guess that I’m premature, but his argument using the Hilbert Hotel is simply ridiculous.

  • My mind is burning! 🙂
    That said, WLC should be shocked at knowing that several centuries ago, a much more famous theologian also believed Universe cannot be infinitely old, yet getting to a point our apologist friend wouldn’t accept easily!
    “Nothing can produce anything before being; therefore, as soon as God was, He created the Universe” (Meister Eckhart, XIII cent.).
    No wonder Eckhart’s ideas were denounced by the Holy See, though he died for natural causes shortly thereafter, without any “help” from the Inquisition.

  • Pofarmer

    It seems to me that even an infinite series had to start somewhere?

    • smrnda

      A funny and counterintuitive fact on infinity.

      Here is the set of integers


      And here are perfect squares


      Because you can map every perfect square to an integer, even though most numbers are not perfect squares, there are the same number of integers and perfect squares.

      Infinity is weird like that.

      • curtcameron

        Even more weird (IMO) is that the number of rational numbers (1/3, 113/355, 98792687624/98798682734) is the same as the integers.

        • curtcameron

          OK, why did Discus put some guy’s picture next to my post? Don’t know who that is, but it’s not me.

        • It shows a cross in the background. That might be annoying for some of us.

        • hector_jones

          What’s wrong with crosses? I find them very useful for hanging my hat. What have you got against hats, Bob? Hmm?

        • At the Atheist Alliance of America conference this weekend, we ate babies. And we liked it.

          You think my black heart cares much for your hats?

        • smrnda

          I see the picture too. That’s a new disqus bug.

          And yeah, infinity is a fun thing. None of these things really make sense, but hey, once you put elements of 2 sets into a one to one correspondence, they’re equal in cardinality.

        • Pofarmer

          Why does disqus do anything? Why doesn’t show me all the comments on a thread that’s still on a single page. Poece of trash.

        • Greg G.

          Why does Disqus post what I type instead of what I meant to type, dammit?

        • Greg G.

          Have you tried to change it by editing your profile?

        • Greg G.

          I don’t think you can say that there are the same number. Infinity is an undefined number but if we add some partial definition, we can do math with it. In smrnda’s example, there is a one-to-one correspondence. In your example, there are an infinite number of rational numbers for every integer in the numerator and an infinite number of rational numbers for every integer in the denominator.

          There are an undefined number of positive odd integers and an undefined number of positive even integers. There is an undefined number of positive integers which is equal to the total number of positive odd and even integers. We can subtract the number of positive odd integers from the number of positive integers and get the number of even integers.

          We had some free time in my Differential Equations class so the professor showed us how he derived an equation that added and multiplied infinite numbers but they all cancelled out at one step and the result was a useful equation for the project he was working on.

        • Compuholic

          You can map every integer to exactly one rational number. That is basically Cantor’s diagonal argument. While it may not be meaningful to talk about the “same number” of integers you can rightfully claim that there are as many rational numbers than integers.

        • Greg G.

          But you cannot map every rational number to an integer so there are more rational numbers than integers. There are integer rational numbers and non-integer rational numbers. If you add the undefined number of each, you get the undefined number of rational numbers.

        • curtcameron

          Actually, you can map every rational number to an integer. That’s exactly what Compuholic was explaining.

          And again, the smirking guy with a cross in the background is not me.

        • Greg G.

          What integer does 3/4 map to?

        • curtcameron

          Depends on how you do it. But the fact that you can do it tells you that the cardinality is the same. In this example, the integer 3/4 maps to is 17.

        • Compuholic

          Like curtcameron said: It depends on how exactly you define the mapping. In my mapping I had the number 16 mapped to 3/4. Take a look at this grid and imagine a line moving back and forth diagonally through the grid. Skip the fractions of equal values ( like 2/2 or 3/3 ) and enumerate all the fractions and there is your mapping.

        • Greg G.

          Aha! It just goes to prove that what Abraham Lincoln said about the internet: “The best way to get correct information is to post incorrect information.”

          After removing the extras that reduce to 1, 1/2, and 2, I get either 13 or 15. Cool!

        • Greg G.

          Perhaps the notation wasn’t clear. “3/4” is supposed to be three fourths, three divided by four, or 0.75. It’s the same as 6/8 and 9/12 and so on, so it maps to any number of integers depending on how you do it.

        • Greg G.

          I didn’t notice the hyperlink the first time I read this. I followed the link from Compuholic and I see the logic, now. Thanks!

        • (Can you go to your Disqus profile and change that photo?)

        • curtcameron

          I wanted to just delete it, but couldn’t find a way to do that in Disqus.

        • You’re Grumpy Goat now? OK. My image is nothing to write home about.

  • Compuholic

    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?

    Yeah, you can ask that. But that would be asking a question about an incoherent concept. It is basically the reverse question of how long does it take to count to infinity? Infinity is not a number so the concept of counting to infinity or counting from infinity are meaningless.

    You might as well ask the following question: “What is colder? At night or outside?

    • Greg G.

      Chuck Norris counted to infinity… twice.

  • smrnda

    The fact that an infinite series could have a finite sum pretty much destroys Craig’s claim, but it may be that people don’t know enough mathematics that his impressive sounding words convince them.

    • Pofarmer

      Well, isn’t what he is doing there just a derivative of Zeno’s paradox, anyway? It makes me think that, once again, philosophy is useless when it dismisses empirical methods. He really should know the answer to this.

      • smrnda

        I’m a big believer in the necessity of empirical methods. I find that areas where people reach bad conclusions (economic policy, for example) is usually because people pay more attention to theoretical models than to empirical data.

        Given how Craig likes to present himself as a polymath, I’d love to see how he would perform on a test of various realms of knowledge.

        • MNb

          “I’m a big believer in the necessity of empirical methods.”
          If you don’t you can’t embrace the scientific method.

  • You can do weird things with infinity.
    If you have a hotel with infinite rooms already all rented out to infinite people and is completely full and one person shows up, you can make room for him by opening up room 1 by moving the person from room one to room 2, room 2 to room 3, room 3 to room 4 and so on and so on. So you can always make room for any more finite number of guests.
    If infinite guests show up though, you can still fit them in by moving the guest currently in room 1 to 2, room 2 to 4, and so on and so forth freeing up all the odd numbered rooms (as their are infinite odd numbers), to fit the newly arrived group of infinite guests.
    In fact you can work out further rules to allow more and more layers, like having infinite cars show up each carrying infinite guests in them. Where things fall apart is if you structure it in a way that you end up uncountably infinite rather than countably infinite which are mathematically quite different.
    But basically, infinity is not very intuitive. It is a lot of fun though!

    • Greg G.

      When you say that the rooms are already rented out, you are saying there are zero empty rooms. By shuffling tenants from one room to the next, you simply have that number of displaced tenants.

      A hotel with an infinite number of rooms could accommodate an infinite number of guests and still have an infinite number of rooms.

      • Sven2547

        And yet, if that hotel’s rooms are numbered using positive integers (1,2,3,4,….) then it does not have enough rooms to house a number of guests equivalent to the number of “real numbers” between 0 and 1. Some infinities are greater than other infinities.

      • Interesting observation. Are you saying that Hilbert makes an error when he wants to free up the odd rooms for an infinite number of new people?

        • Greg G.

          It seems to me that it’s all in how the hotel is described. If there’s an infinite number of rooms and an infinite number of guests, there could still be an infinite number of open rooms and an infinite number of guests can be accommodated without displacing anyone. But if your definition implies there are zero empty rooms, you can only accommodate new guests by displacing guests. You can’t invent empty rooms because you’ve defined them as occupied at the beginn8ng.

          There are conceptually an infinite number of degrees Kelvin but you cannot invent a new one by incrementing each one.

          Suppose Hilbert’s Infinite Pipe was filled with a gas. We could add more gas to one end of the pipe by compressing it and allowing the pressure to dissipate on down the line so as not to compress it so much as to start a fusion reaction.

          That seems like a better illustration by using compression instead of shuffling guests always waiting for the next room to be vacated.

          Edit: Maybe I should have called it Halliburton’s Infinite Pipe.

        • Maybe the conclusion is that Hilbert’s Hotel is not helpful for demonstrating much. I’m not even sure what WLC uses it to illustrate. If his point is simply that infinities are counter-intuitive, that’s true, but not especially helpful.

  • RichardSRussell

    For a little more depth on this subject — but still accessible to anyone with high-school math — I recommend Jordan Ellenberg’s recent book How Not To Be Wrong: The Power of Mathematical Thinking.

  • Pofarmer

    My wifes family is officially fuckin nuts. My middle boy stayed with her sister for a week. They left from a local campground and they all blessed themselves with holy water before they left. Some friends borrowed the camper and had 3 flat tires wherever they were going. My SIL says, “Oh, heck, I forgot to bless it before they left.” I don’t know how I am still married. Much more influence from my wifes family and I’m afraid I wouldn’t be.

    • Pofarmer

      Oh, geez. My SIL’s family has one of those conversion RV’s built on a one ton Ford Van Chassis. So, anyway, some of their friends take it on a golfing trip 100 miles. Blow out an inner dual, put the spare on, which is junk, then blow out both duals when the spare goes. So my SIL’s says “Man, I feel guilty, I forgot to bless it. I blessed it before we went camping and we didn’t have any flat tires.” And all I can think is WTF universe are these people living in? My boys are astute enough to roll their eyes and ignore it, but I am really sorry for their kids.