In the comments section of Bob Murphy’s blog, I was asked (in effect) why I insist on the objective reality of the natural numbers (that is, the counting numbers 0,1,2,3…) but not of, say, the real numbers (that is, the numbers we use to represent lengths — and that are themselves represented by possibly infinite decimal expansions).
There seem to be two kinds of people in the world: Those with enough techncal backgroud that they already know the answer, and those with less technical background, who have no hope — at least without a lot of work — of grasping the answer. I’m going to attempt to bridge that gap here. That means I’m going to throw a certain amount of precision to the winds, in hopes of being comprehensible to a wider audience.
As you might have heard, the set of real numbers is larger than the set of natural numbers, in the sense that there is no one-to-one correspondence between the two sets, and any attempt to construct such a correspondence will leave out some (in fact infinitely many) of the real numbers. We express this by saying that the reals are uncountable.
Now let’s be fanciful. Suppose the Devil comes along and prunes our mathematical universe. He throws away a whole lot of numbers, a whole lot of sets, and a whole lot of functions. When I say he “throws these away”, I mean he somehow arranges that these things will be erased from our mathematics; they will play no role in any calculations or proofs, and, though we will go on merrily doing mathematics as always, we will never notice they are missing.
In fact, he throws away so many real numbers that the remaining ones — the ones we have to work with — form a countable set.
The Devil’s problem is that we might notice that the reals are coutable, say by discovering an explicit one-to-one correspondence with the natural numbers. He solves this problem by insuring that, while he’s busy throwing away all those real numbers, he also throws away all the one-to-one correspondences between the natural numbers and the remaining reals. Those one-to-one correspondences exist, but they’ve been removed from our mathematics so we can’t discover them.
Then we’ll go on merrily believing that the reals are uncountable (in fact, we’ll still have our perfectly good proof of that fact) and there’s a sense in which we will be right. Indeed, when we say “the reals are uncountable”, what that means is: “There is no one-to-one correspondence between the natural numbers and the reals”. And in fact there is no such one-to-one correspondence. There used to be, but the Devil destroyed it.
But although we believe correctly that our set of reals is uncountable, the Devil believes correctly that it’s countable — because he kept copies of all those one-to-one correspondences he threw away. And though we believe correctly that our “set of all real numbers” is the set of all real numbers, the Devil believes correctly otherwise — because he kept copies of all those real numbers that he erased from our Universe.
Now comes the theorem: No matter what mathematical Universe you live in — that is, no matter what sets, functions, numbers, etc. you have access to — there is always a larger mathematical Universe, with more real numbers, and more one-to-one correspondences, whose denizens will believe correctly that our Universe was constructed by a Devil who started with the “true” real numbers, threw most of them away, left us only a countable set of them, and threw away enough one-to-one correspondences to keep us oblivious to that fact. (Of course, the Devil must worry that we’ll somehow discover that those one-to-one correspondences are missing, which means he’ll have to throw away a bunch of other stuff too, to prevent that discovery. And so on. But it turns out that he can always complete this herculean task.)
As we pass to larger and larger mathematical Universes, the set of natural numbers remains the same, while the set of real numbers keeps growing. No matter what Universe you inhabit, the denizens of the “next Universe up” will always agree that you’ve got the natural numbers exactly right, but they’ll sneer at your set of real numbers, which looks to them like a product of the Devil’s work — a mere countable set that doesn’t begin to account for all the “true” reals. Of course, there’s always another Universe where people are saying exactly the same things about them.
In that sense, there is only one true set of natural numbers, but there is no such thing as the “one true set” of real numbers. That makes it easy for me to believe that the natural numbers have a more solid kind of existience than those slippery reals.
That’s the main post. Now let me say a few things by way of partial penance for my (intentional) misprecision. What follows will be (slightly) less imprecise, but if you want true precision you should of course turn to the textbooks. (Try scouring the indexes for terms like forcing and Levy collapse.)
The standard axioms for arithmetic have many models — that is, there are many “number systems” that satisfy those axioms. The standard axioms for set theory also have many models — that is, there are many “mathematical Universes” that satisfy those axioms. The usual tools of first-order logic don’t allow us to distinguish among these models. On the other hand, almost all mathematicians believe that among the many models for arithmetic, there is exactly one, the so-called “standard model” that we’re actually talking about when we talk about arithmetic. (You can use the tools of second-order logic to specify this model uniquely, but I’d argue that that’s cheating. But while we might disagree about how to specify it, we pretty much all agree that the standard model exists.) On the other hand, when it comes to set theory, there is no clear way to point to one of the many models and say “that’s the one I’m talking about”. They all — or at least many of them — seem equally good.
In that sense, when we do arithemetic, we know exactly what we’re talking about. We’re talking about the good old standard natural numbers, and we pretty much all agree on exactly what those are (though we might disagree on how we know what those are). But when we do set theory, we’re in far murkier territory. I might be talking about one Universe, you might be talking about another, and we’d never know it. I’d say “the reals are uncountable” and you’d say “I agree”. But for all we know, we’re talking about different sets of reals, and either of us, if we knew what set the other was talking about, might say “but those aren’t the reals — and dammit, your set is countable”.
If you’ve got a solid advanced undergraduate math background, you might object that the reals are constructed from the naturals. We start with the naturals, form quotients, to get the rational numbers, then proceed to the reals by a process that involves taking limits (or something of the sort). Therefore, you might say, if I know what the naturals are, then I know what the reals are a fortiori. But the reals you construct depend not just on the rational numbers you’ve got available (which are the same in every Universe); they depend also on the sets of rationals you’ve got available — and the available sets differ from one Universe to another. No matter what Universe you live in, your construction of the reals at some point invokes the notion of “all” subsets of the rationals. And somewhere, in a higher Universe, the Devil is chortling — because he knows, and you don’t, that you’re missing a whole lot of subsets, and hence missing a whole lot of reals. But he too is missing a whole lot of reals, according to the next higher Devil.
I’m tempted to sum this up by saying that the natural numbers are real, but the real numbers are imaginary. Or, as Kronecker put it, “God created the natural numbers; all else is the work of Man”. (Note to Bob Murphy: My endorsement of Kronecker is contingent on a sufficiently metaphorical interpretation of the word “God”.)