Topics Mathematics

Question Can something be infinite if there is a definitive number of it? Here's an example: I take a number, the largest I can think of, and never stop adding one to it. The number becomes infinite. Now if you take the number of human beings, and never stop adding to it, is the number of human beings infinite? In contrast, dinosaurs cannot be added to therefore they would not be infinite. Does this make sense?

Accepted:
December 2, 2005

Comments

If there are exactly 5 chairs in my apartment, then the number of chairs in my apartment is not infinite.

But what if, in terms of those chairs, I could generate a sixth, and in terms of those a seventh, and so on, without end? (Of course, that is false. I'm assuming that's why you chose humans as your example, because you're imagining that given any collection of human beings we could "generate" another -- I don't think that's true as a matter of fact either, but let's not get hung up on the example.)

Some mathematicians (they are sometimes called "intuitionists") would say that under those circumstances there would be an infinite number of chairs. It would be true to say that, while at any given time there are actually only a finite number of chairs, still the collection of chairs is infinite. Its infinitude doesn't consist in the fact that there are actually an infinite number of chairs (for there never is actually more than a finite number of chairs), but rather in the fact that we can keep on generating more chairs forever.

For more on this conception of infinity and its contrast with a more prevalent view, see Question 139.

There's a part of your question that I think requires clarification. You say that if you keep adding one to a number, then the number "becomes infinite". I don't think I would say that. The number keeps increasing, and it will eventually exceed one million, or one billion, or any other number I might choose. But it is always finite; it never actually becomes infinite.

Similarly, if you keep generating people (or chairs, in Alex's example), then the number of people keeps increasing, but it is always finite. (As Alex said in his example, "at any given time there are actually only a finite number of chairs.") However, you could talk about the collection of all people ever generated by this (infinite) process, and that would be an infinite collection.

Thus, it is important to distinguish between the collection of all people in existence at any particular time, and the collection of all people ever generated by the process. The former is always finite, but the latter is infinite. (By the way, not only intuitionists but also classical mathematicians would say this, although they would mean somewhat different things by it.)