How Infinitely Big Is God?

Perhaps smaller than a fraction of a fraction. /

In the history of mathematics, recreational studies have often led to sublime discoveries.

Pascal and Fermat were contemplating how to cheat in a game of chance, to “determine if certain outcomes are predictable,” when they developed our modern understanding of probability.

Famously, Archimedes was bathing when he formulated the principle of displacement (“Eureka!”).

So it’s not out of the question to imagine that maybe mathematics help us worship God. In fact, Set Theory helps do just that.

How Big Is Infinity?

Set Theory is a fancy name for a simple idea: “Let’s study things in a collection!” For mathematicians, these things often include mathematical topics such as numbers, equations, shapes, and so forth.

You can use Set Theory to study the cereal aisle at a grocery store. What cereal boxes have a cartoon character on them (Set A)? What cereals will Mom let me eat for breakfast (Set B)? Does Set A or Set B have more objects? (Probably Set A.) Is there any overlap between Set A and Set B? (Of course not.) And so on.

Set Theory can become very complex, but at its core, it’s straightforward enough. You may not realize that it was highly controversial when it was first introduced in the late 1800s. Georg Cantor, the mathematician responsible for developing it, received the label “Corrupter of Youth” from his contemporary Leopold Kronecker. Mathematician Henri Poincaré called Cantor’s work a “grave disease” that infected mathematics. Cantor’s work received the same reaction that church moms gave KISS in the 1970s and ’80s.

What could have riled up Kronecker and Poincaré to offer such strident criticism of Cantor’s work? What appears innocuous has startling ramifications when examined more closely.

Cantor classified the set of all counting numbers (1, 2, 3, 4 . . .) as an infinite set. There’s no end to that list; there are countless counting numbers. This is a widely accepted fact about counting numbers, and was generally accepted at the time.

But when Cantor looked at the set of all real numbers (all the counting numbers and all the negative numbers and all the decimals and fractions between those positive and negative numbers), he concluded that real numbers were “more numerous” than counting numbers.

Did you catch the problem? If the set of counting numbers is never-ending—that is, infinite in size—how can another set contain more numbers than that? The basic assumption had been that all infinite sets contained the same innumerable quantity of objects.

Cantor was not merely making a guess or assumption; he offered a mathematical proof that there exist an “infinity of infinities.” That is, there are more infinities than we can count.

Moreover, he argued, if other infinities exist, then we must be able to “do math” with them. Addition, subtraction, multiplication, division, exponents, and roots—all must be fair game if there are multiple (indeed, uncountable) infinities.

Mathematicians like Poincaré didn’t want to consider multiple infinities, as this idea had the potential to erode the certainty of computational mathematics. Ludwig Wittgenstein criticized Cantor’s discoveries as “pernicious idioms” since they challenged his mathematical philosophy, summed up as, “We can create our own understanding of mathematics.”

The response from the Christian community was no different: This idea of “multiple infinities” smacked of relativism and pluralism, the great enemies of the day’s orthodoxy.

Today, many Christians still look askance at Cantor’s discoveries. A popular homeschool textbook sponsored by a prominent Christian university proudly claims that its math curriculum “provides attractive, legible, and workable traditional mathematics texts that are not burdened with modern theories such as Set Theory.” Well, then.

The God of Multiple Infinities

What if instead of threatening our theology, Cantor’s discovery of multiple infinities actually enhanced our understanding of God?

A number of biblical verses use spatial or numerical metaphors to help us grasp the infinity of grace as well as of God himself. Like, “[A]s far as the east is from the west, so far has he removed our transgressions from us” (Ps. 103:12). How far is the east from the west exactly? Sounds like infinity to me. “‘I am the Alpha and the Omega,’ says the Lord God, ‘who is, and who was, and who is to come’” (Rev. 1:8). When did God “begin”? How long is eternity? What is time like for God? Infinity seems like a part of the answer to each of these questions.

But there is one verse for which Set Theory seems especially apt: “Are not two sparrows sold for a penny? Yet not one of them will fall to the ground outside your Father’s care. And even the very hairs of your head are all numbered. So don’t be afraid; you are worth more than many sparrows” (Matt. 10:29–31).

When most of us think of “infinity,” we think of numbers climbing toward some enormous, unreachable total. But again, note Cantor’s discovery. The set of real numbers is not a larger infinite because those numbers go “farther into infinity.” It’s the spaces between the counting numbers that give the set of real numbers its size. It’s in the minute detail, the countless rational and irrational numbers squeezed between each other and between the counting numbers: The square root of two, –1/3, Pi. Cantor demonstrated that there are more real numbers between zero and one than there are counting numbers between one and infinity.

Similarly, God’s infinitude is not found only in his massive, eternal, overwhelming presence, but in every minute detail of life, every crack and crevice of the universe. So much so that he knows the number of hairs on our head and grains of sand on the beach, and he counts every star in the heavens, each of which—by the way—he also knows by name (Ps. 147:4).

God is not only an infinitely distant and big God, but also infinitely present in the smallest details.

Stupefied Praise

One other verse worth noting: “Oh, the depth of the riches of the wisdom and knowledge of God! How unsearchable his judgments, and his paths beyond tracing out!” (Rom. 11:33). Cantor, himself a devout Christian, believed that he was called to reveal Set Theory to the world, and that it at least suggested the reality of God. He wrote, “There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact, the necessity of the actual infinite can be demonstrated” (emphasis added).

No one argues that Set Theory proves God’s existence or anything about his character. But those with eyes of faith nonetheless see a reflection of God in it, even as the theory and God leave us finally speechless: His ways are even more inscrutable than Set Theory. In the face of such mystery, I often feel compelled to simply offer praise.

Joel Bezaire is a teacher and musician from Nashville, Tennessee. He teaches at the University School of Nashville and attends West End Community Church with his wife, Jennifer, and two young boys. Follow him on Twitter @joelbezaire and connect with him at

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Also in this Issue

Issue 11 / December 11, 2014
  1. Editors’ Note
  2. Tiny Creatures of Great Worth

    Bacteria: there are trillions of them. And many are on our side. /

  3. All Streams Lead to Christ

    The splendor of God’s revelation is that it is both manifold and one. /

  4. Gaudete

    Which is to say, rejoice! /

  5. Wonder on the Web

    Links to amazing stuff

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