Why Infinity Is Not Normal

The following is excerpted from the book Beyond Infinity: An Expedition to the Outer Limits of Mathematics, by Eugenia Cheng. In this very interesting excerpt Cheng, who is an Honorary Fellow in the School of Mathematics and Statistics at the University of Sheffield, U.K., and is Scientist in Residence at the School of the Art Institute of Chicago, explains why infinity is not a number in the ordinary sense and why we have to be careful in how we talk about it.

The passage begins with Cheng asking us to think about what we mean when we talk about infinity:

Mathematics is all about using logic to understand things, and we’ll find that if we’re not careful about exactly what we mean by “infinity,” then logic will take us to some very strange places that we didn’t intend to go....In the previous chapter I listed some beginning ideas about infinity.

Infinity goes on forever.

Does this mean infinity is a type of time, or space? A length?

Infinity is bigger than the biggest number.

Infinity is bigger than anything we can think of.

Now infinity seems to be a type of size. Or is it something more abstract: a number, which we can then use to measure time, space, length, size, and indeed anything we want? Our next thoughts seem to treat infinity as if it is in fact a number.

But if we treat infinity like a normal number we get contradictions:

If you add one to infinity it’s still infinity. This is saying

∞ + 1 = ∞

which might seem like a very basic principle about infinity. If infinity is the biggest thing there is, then adding one can’t make it any bigger. Or can it? What if we then subtract infinity from both sides? If we use some familiar rules of cancellation, this will just get rid of the infinity on each side, leaving

1 = 0

which is a disaster. Something has evidently gone wrong. The next thought makes more things go wrong:

If you add infinity to infinity it’s still infinity. This seems to be saying

∞ + ∞ = ∞

that is,

2∞ = ∞

and now if we divide both sides by infinity this might look like we can just cancel out the infinity on each side, leaving

2 = 1

which is another disaster. Maybe you can now guess that something terrible will happen if we think too hard about the last idea:

If you multiply infinity by infinity it’s still infinity. If we write this out we get

∞ x ∞ = ∞

and if we divide both sides by infinity, canceling out one infinity on each side, we get

∞ = 1

which is possibly the worst, most wrong outcome of them all. Infinity is supposed to be the biggest thing there is; it is definitely not supposed to be equal to something as small as 1.

What has gone wrong? The problem is that we have manipulated equations as if infinity were an ordinary number, without knowing if it is or not. One of the first things we’re going to see in this book is what infinity isn’t, and it definitely isn’t an ordinary number. We are gradually going to work our way toward finding what type of “thing” it makes sense for infinity to be.

You can read more of Cheng's thoughts on infinity here.

Sometimes scientists trying to avoid the fine-tuning problem or an initial origin event of the cosmos say things like there's an infinite number of universes in the multiverse or that the cosmos is infinitely old. Cheng shows that we have to be very careful about such uses of the word.

In fact, one argument against the universe being infinitely old is that if it is infinitely old then there has been an infinite number of moments of time. But if so, then there was no first moment, because if time is infinite in the past whichever moment one designates as "first" will always have been preceded by an earlier moment, and, if there was no first moment there could have been no second, or third moment, etc. The consequence of this is that if there were no first, second, third etc. moments then we could never have arrived at the present moment. But, of course, we have arrived at the present moment, which means that the universe must not be infinitely old. It must have had a beginning, a first moment.

This, then, provokes the question, "If the universe had a beginning, what caused it?" Whatever the cause, it must have been outside of space, outside of time (because these are components of the universe), very powerful and very intelligent. In other words, the cause must have been something like God.