Infinity By Steve Humphrey

Infinity? What a concept! Many use the term unthinkingly, but do people really know what they are talking about? I think not.

There are several topics of discussion to which the concept of infinity is relevant, but perhaps the easiest to understand is the realm of numbers. There are different kinds of numbers. Counting numbers (1, 2, 3, …), also called positive integers. Zero. Negative numbers (integers less than zero, e.g., -1, -2, -3, …). Rational numbers, or fractions, can be expressed as the ratio between two integers (1/2, 1/3, 2/3, 5/6, etc.). Non-rational reals, which cannot be expressed as ratios (e.g., pi and √2). How many of each of these are there? Well, infinite, but there are different kinds of infinity involved.

One intuitive notion of infinity is unbounded, having no end. The counting numbers have a first element, but no last element. No matter what number you specify, you can always add one to get a larger number, so they are infinite in the positive direction but not in the other direction. The set of all integers, positive, negative and zero, is infinite in both directions, in the sense that there is a simple mechanism for extending the sequence in both directions. But now, consider the set of fractions between zero and one. This set is clearly bounded on both sides, but it is still infinite (1/2, 2/3, 3/4, 4/5, 5/6, 6/7, …).

And when it comes to numbers, our intuitions are quickly violated. It turns out that there are the same number of counting numbers as there are fractions between zero and one, and in fact, the same number of integers as there are fractions of all sizes. But how can this be? If there are an infinite number of fractions between zero and one, and an infinite number between one and two, etc., how can there be the same number of counting numbers as there are fractions? There is a simple proof due to Georg Cantor that demonstrates this, which I may elaborate on in a future column. To make matters worse, there is a similar proof that shows that there are more non-rational real numbers than there are integers. That is, though both sets are infinite, the set of reals is bigger than the set of counting numbers or fractions. And, in fact, there is an infinite sequence of ever larger infinite sets. But how can something be “more infinite”?

Now, we can use these notions of infinity to talk about the physical world, rather than just numbers. Consider time. Many people believe that the Universe had a beginning, either through Divine Creation or a Big Bang, so perhaps there was a first moment. How about a last moment? Is the Universe eternal? Ever since Edwin Hubble discovered that the Universe is expanding, there has been debate over the ultimate fate of the physical world, depending upon how much mass there is and how fast it was expanding at the beginning. If there is sufficient mass, expansion should slow down due to gravitational attraction, and either come to a stop and stay there, or recollapse into a Big Crunch, which would represent an end to time. In the late 1990s, astronomers discovered that the expansion of the Universe is accelerating, i.e., not slowing down at all. This discovery is not uncontroversial, and there is no explanation for it (though we do have a name for whatever it is that accounts for it: Dark Energy). If it is true, then the Universe should last forever. Maybe. If the Universe becomes cold and dark, and even sub-atomic particles become widely separated, why would we say that time is passing? (I talked about this in an earlier column.) If the Universe ends “not with a bang, but with a whimper,” then even though there may be no last moment, it might still not be eternal. Maybe the model is the fractions less than one. There isn’t a last one, but they don’t go past one.

Similar things can be said about space. Is the Universe infinitely big? Is it like counting numbers, in that no matter how far you go, you can always go a bit farther? Or would you come to an edge, like pre-Columbian sailors believed they would reach in sailing across the Atlantic? Again, this is tough to answer. Consider the surface of the globe. Leaving aside political boundaries, deep oceans and tall mountains, there really isn’t an edge, or end. You can travel around and around the Earth without ever reaching a limit. But the Earth is not infinitely large, even though it has no edge. It is finite but unbounded. The Universe may have a geometrical structure akin to that of a globe, where you could go and go and never reach the end, and maybe end up where you started. A fanciful way of illustrating this is to say that if you looked long enough through a powerful enough telescope, you might see the back of your head!

So, I think the lesson here is that while mathematicians have provided a rigorous account of the concept of infinity, applying it to questions of real, physical infinities proves to be quite difficult. And, going back to our initial intuitive notion, since we’ve seen examples of things that are infinite and bounded, and things that are finite and unbounded, “infinite” cannot mean “unbounded.”

Steve Humphrey has a Ph.D. in the history and philosophy of science, with a specialty in the philosophy of physics. Questions? Comments? Suggestions? Email him at steve@thevoicelouisville.com.