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To Infinity… and Beyond!

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It was probably the craziest children’s sermon intro I will ever try. “What’s this?” I asked as I showed them a sideways eight or ∞.

I had expected to get a cute reply like “a tired 8, laying down for a rest.” Instead I got blank looks. Finally, one kid asked, “Is it an 8?” “No,” I replied, rotating the sign to show the difference between an 8 and ∞.

Then I told them it was the symbol for infinity. I asked them about infinity—this was after Buzz Lightyear’s famous line—so they had some idea of what it might mean. They suggested something “really big.” So, we worked with that idea; we talked some about really big numbers, about the nearly incomprehensible size and age of the universe, and then made the connection to the infinite God who created it all.

I’m not sure if that message landed—it may have more with the adults than the kids—but I have always been intrigued by mathematical and scientific work on infinity. I’ve wondered how to connect it with Christian belief in the God whom we often call “infinite.”

In Math and in Nature?

The basic theological concept is pretty simple. God is infinite (or “beyond measure”) according to Psalm 147 (see also Job 11, Isaiah 55 and Romans 11). Creation, on the other hand, including “the grass,” “the flowers,” and “humankind,” is finite (Psalm 103). Some say sin has something to do with our finitude, but I suspect that all creation, with or without sin, is finite.

Like the theological concept, the mathematical one starts out simple enough. Infinity is like a list that never ends, such as the number of numbers you could count if you could count eternally. Or, consider a puzzle from the ancient Greek philosopher Zeno, who argued that you could never fully travel the distance between two points. To do so, you first have to travel half of it, and then half of the remaining distance, and so on. Just imagine going half way ad infinitum—you will get exceedingly close, but never arrive!

While the concept starts off simple enough, infinity gets pretty complicated. Think about the sets of even and odd numbers. It seems obvious these sets are the same size, and we can match up every odd number with exactly one even number: 1 with 2, 3 with 4, 5 with 6, and so on. Now, clearly, if we put the even and odd numbers together, we get a bigger set, right? Not so fast, say the mathematicians. Because these sets are infinite, we can still match up every number in one set with exactly one number in the other: 1 with 2, 2 with 4, 3 with 6, 4 with 8, and so on.

So, the set of all whole numbers is the same infinite size as the set of even numbers—and we start to see why infinite sets are strange. Indeed, any set that we can put into an ordered list shares the same infinite size as the whole numbers. (I’m told even the set of all fractions can be put into an ordered list!) Mathematicians call such sets countably infinite.

This raises the question, then, of whether all infinities are countable. Enter Georg Cantor, a German mathematician who developed much of our modern understanding of infinite sets. Cantor also studied the real numbers—all the numbers that you can write in decimal form, including things like pi and the square root of two. In a famous theorem, Cantor was able to prove that the real numbers are not countably infinite. Essentially, he shows us that no matter how you try to put the real numbers into an ordered list (so that you could match them up with the whole numbers), you will always be able to find another real number that doesn’t fit in your list. So it seems some infinities are indeed larger than others.

The next big question, then, is do infinities actually exist in nature? Sure we can imagine the possibility of never ending lists, but do such entities map onto to something real in nature?  Perhaps it’s a black hole, or the initial singularity of the universe, or other places where our physics breaks down.

Well, the answer to the question of actual infinities in nature is that we don’t know. Mathematicians and physicists have opinions, but no one has ever seen or measured an infinity. Check out some of the links below for additional perspectives.

Qualitatively Infinite

I began by suggesting that the notion of infinity was simple—God is infinite and we are not.  But is it simple? What do we mean when we refer to God as infinite?

We often use it to describe divine transcendence—God is not merely immanent, inside of (what appears to be) the finite space and time where creatures live and move and have our being. Sometimes it references the size of God: it would take something like Anselm’s God, “a being than which no greater can be conceived,” to create a universe with complex things like our brains in a universe billions of years old and billions of light years in volume, not to mention the possibility of an uncountable multiverse with an uncountable number of beings. A God that big is worthy of being called infinite.

But perhaps it applies best to God’s attributes—things like God’s love, knowledge, even creativity. And rather than the quantities we summarized above, I like to think about God’s infinity in qualitative terms.  God is infinite love and mercy and knowledge not in terms of anything countable, but in terms of a quality of perfection. Cantor, a devout Lutheran, called this Absolute Infinity which he differentiated from creaturely infinities, or the transfinite.

That is to say, as much fun as this foray into the concept of infinity in mathematics and science has been, for it to land, we may have to follow Buzz Lightyear to infinity and beyond.  God is beyond uncountable lists—even the largest of all possible infinities (whatever that means)—and by infinite, we imply something like God’s perfect being. What distinguishes the Infinite from the finitude of creation is God’s perfection—which dwelt among us in the finite person of Jesus and remains with us finite creatures through the Holy Spirit.

Cheers,

Drew

P.S. A big thanks to mathematician and seminarian Chris Micklewright for help with this edition.

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