Brackets and Air

Are numbers the only thing that can be infinite?

When we say that numbers are infinite, we mean that in counting we proceed like so: “1, 2, 3…” with the understanding that in principle we can continue indefinitely. In writing we use points of ellipsis to represent this principle, while in speech we use a phrase like “and so forth”. Expressions like these are just markers for the concept of infinity, its synonyms. They don’t actually explain what it means. Infinity is a difficult concept to get your head around.

We’re most likely to be forced to wrangle with the concept of infinity when contemplating either space or time. Space seems infinite because where would it end, a giant wall? (And if it did we could still ask, what’s on the other side?) Time seems infinite because otherwise what happens, do things just stop? (And if they do we could still ask, for how long do they stop?) But try to imagine infinite space. Maybe picture a cubic centimeter of interstellar vacuum, and another one next to that, and another one next to that, and so forth. But you can’t actually imagine these things. What you do is fix in your head an image of inky blackness with some stars in the background, and then you think, “Surely that can be divided up into cubic centimeters.” You can’t hold even a finite number of these locations in your head at once because they are all indistinguishable.

(One way to handle the infinite space scenario is to pick a reference point and describe other points relative to it. This point is such-and-such a distance from the reference. This point is some other distance. And so on. But a phrase like “such-and-such” is just a placeholder for a number. We cheat by talking about numbers in the guise of talking about the physical world.)

Maybe it would be easier to imagine an infinite number of easily distinguishable things, like people. Say the universe is infinite, and tucked away in its recesses are an infinite number of Earth-like planets inhabited by other human beings. This is not inconceivable. But try to imagine an infinite number of people and you will also fail. You’ll call up a montage of friends, celebrities, and people you pass on the street–twenty or thirty, tops–then say to yourself, “And more like them.” But can you say with confidence that you know what this means?  A wide variety of properties go into characterizing human beings. What if some of those properties cannot be extended indefinitely? How would we ever know?

If there’s a difference between the points in outer space and the human beings, it’s that the former are too simple to be considered en masse, and the latter are too complex to be multiplied without end.

At first glance, the situation with numbers appears similar. You can’t imagine an infinite collection of numbers. It’s difficult to even maintain a clear concept of a single relatively small number. (Can you picture 17,383 unique things in a way that is meaningfully distinct from 16,256 things?) There’s still that ellipsis at the end of “1,2,3…” But in mathematics things are not as hopeless. Arguably the taming of that ellipsis has been part of the project all along, and that goal became explicit in the latter half of the nineteenth century, particularly in the work of Georg Cantor. By expressing the concept of number in terms of the more primitive concept of set, and the act of counting in terms of a successor function that transforms one set into another in a very basic and mechanical manner, mathematicians have been able to gain traction on the idea of infinity, to characterize its properties and consequences with something more than allusions to grandiose vagueness.

Here’s part of how it works. Define the number zero as the empty set {}. Define the successor of any number greater than zero as the set of all numbers that precede it. So 0 = {}, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}, where the Arabic numerals are just convenient shorthand for certain kinds of sets that can be written more explicitly like so:

0 {}
1 {{}}
2 {{}, {{}}}
3 {{}, {{}}, {{}, {{}}}}

If you aren’t familiar with set theory, this might look like it makes counting a lot harder. There isn’t the space here to demonstrate why this is not the case, so you’ll just have to take my word for it that the right hand column above does indeed provide a simpler concept of number that makes the ellipsis at its foot something other than just a placeholder for the limits of our imagination. (If you want the whole story, P.R. Halmos’ Naive Set Theory is probably the place to start.)

Even if you don’t understand set theory, however, the picture above gives a rough sense of the complexity of the concept of a number. Specifically, it illustrates that complexity relative to a human being, or a point in outer space. Notice that, unlike human beings, numbers are extremely simple things: just a bunch empty sets nested inside each other in a certain pattern. The whole of the integers is just brackets and air. And yet, unlike the cubic centimeters in outer space, the numbers are not indistinguishable. The particular pattern of nesting ensures that 1 looks different from 2 and 3. The rule that leads us from a given number to its successor, though simple, provides enough structure to ensure we’ll always be able to tell all its outcomes apart.

(As an example of an insufficiently complex successor rule, imagine instead one that instead defined a number’s successor to be to union of that number with the empty set. Then 0 = {} = 1 = {} ∪ {} = 2 = {} ∪ {} ∪ {} = 3 = {} ∪ {} ∪ {} ∪ {} = 4. This would be useless as the foundation of a counting system, because all the numbers would look the same.)

Infinity is a difficult property to possess, but numbers may posses it by virtue of occupying a sweet spot of having just enough structure to be distinct while remaining unencumbered by particulars. This is a difficult needle to thread, but they manage. Numbers are the only thing that can be infinite.

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One Response to Brackets and Air

  1. Pingback: Baggage | Corner Cases

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