Topics Mathematics

Question Is infinity a number or not and why?

Accepted:
January 29, 2006

Comments

In a sense, the answer, strictly speaking, is "no". Infinity isn't a number. It's a property of sets. Some sets are infinite; some are not. But in a more interesting sense, the answer is "yes": There are infinite numbers. There are many ways to see this. Here's one, borrowed from Frege.

What do we mean by a "number"? Well, a number is the kind of thing one can give as an answer to a question like, "How many books/dolls/cars/whatever are there?" (These are so-called "cardinal" numbers.) There are lots of such numbers, and some of them we call "finite". These are the ordinary natural numbers, zero, one, two, and so forth. Now, notice the following. Every natural number is the number of natural numbers less than it. Thus, each natural number is one less than the number of natural numbers less than or equal to it, which is to say that each natural number is strictly less than—that is, less than and not equal to—the number of numbers less than or equal to it.

Now: How many natural numbers are there? If there is an answer to this question, then it cannot be any finite (that is, natural) number. Why not? Well, suppose it was the natural number n. Then since n is a natural number, n is strictly less than the number of natural numbers less than or equal to n. But every natural number less than or equal to n is a natural number (duh), so the number of natural numbers must be at least that big, and that is, by the preceding, strictly greater than n. So n is not the number of natural numbers, after all. Or, to put it differently, let n be a natural number. It is strictly less than the number of natural numbers less than or equal to it; but the number of natural numbers obviously must be greater than or equal to the number of natural numbers less than or equal to n, so it is greater than n. But n was arbitrary, and so the number of natural numbers is greater than every natural number.

You might therefore suppose that the question how many natural numbers there are could not have an intelligible answer, and for a long time many mathematicians, even some very good ones, thought just that. The philosopher and mathematician Bernard Bolzano wrote a book in the middle of the nineteenth century, Paradoxes of the Infinite, arguing for that claim. But it turns out that the question can have a sensible answer (we can prove that the supposition that there is such a number will not lead to contradiction), and it does have a sensible answer. The answer is "aleph zero", written ℵ0. Which number is that? It's the number of natural numbers. It has some strange properties. For example, ℵ0 + ℵ0 = ℵ0 and ℵ0 × ℵ0 = ℵ0, too. But one would expect that it would have strange properties. After all, it's infinite.

There are bigger infinite numbers, too. But you have to be careful here. You might think, oh, that's easy. Surely there must be more integers, that is, positive and negative numbers, than there are just positive numbers. But there aren't. The number of integers is also ℵ0, precisely because ℵ0 + ℵ0 = ℵ0. Indeed, the number of rational numbers (fractions) is also ℵ0, even though there are ℵ0 rational numbers between any two natural numbers and even between any two rationals! That's because ℵ0 × ℵ0 = ℵ0.

There are many sets of natural numbers. For example, there is the set {0}, the set {0,1}, the set of evens, the set of odds, the set of multiples of seven, and so forth. How many sets of natural numbers are there? Georg Cantor proved, around 1880, that however many there are, there are more than there are natural numbers: That is, there are more than ℵ0 sets of natural numbers. This number is 2ℵ0, and whether it is the next infinite (cardinal) number after ℵ0—called ℵ1—is one of the great unsolved problems of mathematics. Most mathematicians think it probably isn't—the smart money seems to be on ℵ2—but there's no proof of this claim, and we can also prove that none of the (plausible) mathematical principles that are nowadays seriously considered decide the question one way or the other.

I think it's worth pointing out that there are many different number systems, which mathematicians use for different purposes, so your question is really ambiguous. If you are interested in determining how many things of some particular kind there are, then the appropriate numbers to use are the cardinal numbers, and as Richard has explained, there are indeed infinite cardinal numbers. On the other hand, if you're a student in a calculus class, then the numbers you are using are probably the real numbers, and all of the real numbers are finite. Although the symbol for infinity is used in calculus, it is not used as a name for a real number.

Question Is infinity a number or not and why? Accepted:January 29, 2006

Question Is infinity a number or not and why?

Accepted:
January 29, 2006