A Short Post on Infinity

I was recently asked to write a few paragraphs on the mathematical concept of infinity for a school news letter. I have copied it below. It is indeed brief for the subject that it deals with. I encourage those interested to do additional reading. I especially encourage reading the chapter on infinity in Math through the Eyes of Faith.


Infinity is a difficult concept to grasp. We often misuse the term “infinity” to mean “something really, REALLY, big.” When Buzz Lightyear exclaims, “To infinity, and beyond!” the implication is “Let’s go really, really far… and then past that… I guess.” However, when we say that we worship an infinite God, we must be saying more than simply, “God is really, REALLY, big.” So then what are we saying? We can find some insight in mathematics.

To get an idea of infinity in mathematics we have to first be clear on some basic terms and definitions. We can count the numbers in a set by comparing them to the natural numbers (whole, positive numbers). {3,5,7,9} has 4 numbers because we can match each number in this set to the natural numbers 1 through 4: {1->3,2->5,3->7,4->9}. We say this set has size 4. A finite set is any set that can match to the numbers {1,2,3,…,n} where n is some number. An infinite set is a set that is not finite. In other words, there is no stopping point. The natural numbers themselves are infinite: {1,2,3,…}. They just keep going.

Now for some fun.

Consider the even numbers {2,4,6,8,…}. Also infinite. Half as big as the natural numbers right? Wrong. The set of even numbers, which is the set of natural numbers minus the odd numbers, is actually the SAME SIZE as the natural numbers!!! This seems counterintuitive, but we can match each even to the natural numbers {1->2, 2->4, 3->6, 4->8…}. For any even number you can think of, I can give you the natural number it pairs with. We can also prove that the set of all fractions {1/1, 1/2, 1/3, 1/4, …, 2/1, 2/2, 2/3, ….} which intuitively seems much bigger than the set of natural numbers, is also the SAME SIZE as the natural numbers.

So is every infinite set the same size? Nope. The set of all real numbers (every decimal expansion) is infinite, but LARGER than the set of natural numbers. This was proven by George Cantor in the late 1800’s. In fact, it has been proven there are infinitely many different sizes of infinity! Try wrapping your brain around that. Cantor spent his whole life working with concepts of infinity… and he went insane… seriously.

So when we say that we worship an infinite God, what are we saying? From a math perspective there are some familiar aspects about infinity, but it is also wholly different than anything we have ever encountered. It seems to follow rules of logic, yet it is surprising and mysterious. A lot of the characteristics of God that may seem paradoxical on the surface (transcendent yet immanent, perfectly just and yet perfectly gracious, one and three) may not be so paradoxical when you are talking about the infinite.

I’m not proposing any answers to questions of faith based on mathematics. It is my hope that you will see how studying math may give us just as much opportunity to reflect on the wonder of God as does a beautiful painting, song, or piece of poetry.

Enjoy pondering the infinite!

[After my initial post, I received another great comment from Scott Eberle that I thought would be worth including in the post itself]

Infinity is such a great subject for exploring the impact of our faith on mathematics!

Yes, Cantor suffered from depression and had multiple mental breakdowns, partly because of the intense opposition to his ideas. But what is really interesting to me is the whole reason he pursued the study of infinity to begin with.

Up until Cantor’s time, Aristotle’s idea that “actual infinity” does not exist was generally accepted by everyone. This was Aristotle’s way of avoiding the seeming paradoxes associated with infinity. Aristotle taught that we could accept “potential infinity”—that we could always keep going out as far as we needed—but that a real, “actual infinity” does not exist; we can never “get there.” And because mathematicians could not figure out how to deal with infinite paradoxes (like there being as many even numbers as whole numbers), Aristotle’s ideas were accepted. A few mathematicians, like Bolzano and Galileo, toyed with attempts to study actual infinity, but without modern set theory, they did not get very far.

Cantor, on the other hand, was a devout believer. He knew that God was infinite and that “actual infinity” must really exist. And because of this deep-seated conviction, he passionately pursued the study of infinity and developed set theory to describe infinite sets in the face of much opposition, especially from Kronecker, one of his teachers. Cantor insisted that his pursuit of infinity was founded on the theological premise that infinity was an attribute of God and that it was right for us to study it. Studying infinity, for Cantor, was a call of God.

At the time, many mathematicians rejected Cantor’s work and there was quite a lot of opposition. Today, virtually all mathematicians accept it, and the set theory he developed is today considered the very foundation for all mathematics. A real story of faith.