Topics Mathematics

Question Hi, I was hoping for some clarification from Professor Maitzen about his comments on infinite sets (on March 7). The fact that every natural number has a successor is only true for the natural numbers so far encountered (and imagined, I suppose). Granted, I can't conceive of how it could be that we couldn't just add 1 to any natural number to get another one, but that doesn't mean it's impossible. It seems quite strange, but there are some professional mathematicians who claim that the existence of a largest natural number (probably so large that we would never come close to dealing with it) is much less strange and problematic than many of the conclusions that result from the acceptance of infinities. If we want to define natural numbers such that each natural number by definition has a successor, then mathematical induction tells us there are infinitely many of them. But mathematical induction itself only proves things given certain mathematical definitions. Whether those definitions indeed correspond to reality is another question. Am I missing something here? Thanks so much!!

Accepted:
March 14, 2013

Comments

Prof. Maitzen will. I hope, offer his own response, but I'm a bit puzzled.

First, I'm not sure which professional mathematicians you have in mind, but that's not so important. Let's start elsewhere.

The usual axioms of arithmetic do, indeed, tell us that every natural number has a successor. From that it follows with no need for induction that there's no largest natural number. For suppose N is the largest natural number. Then N+1, its successor, is also a natural number, and is perforce larger than N. So I'm tempted to ask if I'm missing something.

The problem I'm having is that I don't know what I'm being asked to contemplate. Perhaps there's some sense of "possible" (though I'm not convinced), on which it's possible that we're so massively deluded that we can't even get simple arguments like the one just given right. But in that case, all argumentative bets are off. Put another way, if we're wrong in thinking there's no largest natural number, then we're so hopelessly confused that we'd have no reason to trust our reasoning about much of anything. And in that case, we'd have no reason to trust our reasoning about the question you've posed.

As it happens, however, there's no reason to think there's a reason not to trust our reasoning here...

Thanks for sending a follow-up question. Prof. Heck, who knows this territory better than I do, provided helpful corrections and amplifications in his answer to Question 5068. I recommend taking another look there.

You wrote, "The fact that every natural number has a successor is only true for the natural numbers so far encountered (and imagined, I suppose)." The claim that every natural number has a unique successor, like the claim that 1 isn't the successor of any natural number, is an axiom -- a starting point -- rather than a conclusion drawn from examining or imagining data. Your fifth sentence suggests that you know this already. You're quite right that math induction proves its results only given its starting points, but of course that's true of all proofs: all proofs (even proofs that have no premises) rely on essential assumptions.

You say that some professional mathematicians would rather accept the existence of a largest natural number than accept the paradoxical features of infinite collections. I'm no professional mathematician, but I have to say that the paradoxes of infinity that I know about don't bother me so much that I'd go their route. All of those paradoxes have solutions that I can at least imagine being correct, whereas (for what it's worth) I can't even imagine a natural number's having no successor.

I'm not familiar, either, with any working mathematicians who think there is a largest natural number or, more specifically, that there are only finitely many numbers. I do know of some work, by Graham Priest, that investigates finite models of arithmetical theories, but this is in the context of so-called paraconsistent logics. In Priest's theories, it is true that there is a greatest natural number, but it is also true that there isn't one! But that is probably not the kind of thing the questioner meant.

Part of the reason mathematicians are happy with infinity is that infinity is very cheap. Consider, for example, ordered pairs. If you think (a) that, given any two objects, there is an ordered pair of them and (b) that there is an object that is not a pair, then it follows that there are infinitely many pairs. Or consider the English sentences. Not just the ones someone has uttered or written down, since there are ever so many English sentences no one happens to have uttered before (such, I am sure, as the one I am currently writing), but all the English sentences there are. Is there a longest one? Surely not. Is there a longest computer program? Surely not. And so forth.

None of that is to say that there aren't large philosophical questions in this area, such as how exactly we are supposed to know that there is no largest number. There are. And, unsurprisingly, philosophers have disagreed about the answers to such questions. But it is hard to imagine doing any kind of mathematics without being able to assume that every number has a successor, or something more or less equivalent.

It's important to distinguish two different issues: (i) whether there are infinitely many natural numbers; (ii) whether there are mathematical objects that are themselves infinite. The natural numbers are, in some intuitive sense, finite objects. But the set of all natural numbers is, in some sense, an infinite object. And it is possible to accept that there are infinitely many natural numbers without accepting that there is a set of all of them or, more generally, that there are any objects that are, in their own right, infinite. And there are respected mathematicians who hold this kind of view, though they are definitely a minority.

It's also worth distinguishing the question whether there are infinitely many numbers from the question whether every number has a successor. There are natural, and important, theories of arithmetic in which one cannot prove that every natural number has a successor, but in which there are infinitely many such numbers. I explore some of these theories in my paper "Frege Arithmetic and `Everyday Mathematics'", which you can find on my website.