Asaf Karagila
I don't have much choice...

Name that number

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In the best TV show ever produced, Patrick McGoohan plays the mysterious No. Six. He lives in The Village, where former spies are held. The people there are essentially captive, and they all have numbers instead of names. But he is not a number! He is a free man!

We find a similar concept in Zelda's poem "Every man has a name" (לכל איש יש שם), which in Israel is closely associated with the Holocaust and with assigning numbers to people. But alas, we are all numbers in some database. Our ID numbers, employer number, the index under which you appear in the database. You are your phone number, and your bank account number. You are the aggregation of all these numbers. And more.

But we are not here to talk about people, we are here to talk about numbers. Numbers have names as well. \(1\) is the name of the unit, \(\pi\) is the name of the ratio between a circle's circumference and its diameter, and so on.

One common fallacy is that "the set of nameable numbers is countable". Joel Hamkins explains it well. But let me offer a slightly different approach as to why this is an inaccurate state of affairs. And while we're at it, we can investigate what it means for a real number to have a name.

The naive notion of nameable numbers. We like to think about nameable numbers as numbers which we can explicitly define and name. Like \(\pi\), like \(e\). But that is not really hitting the nail on the head. What we do think about is that there is a mathematical definition for the number. So \(\pi\) is some line integral, and \(e\) is a certain limit, or defined from \(\exp\) (which in turn can be defined as a unique function satisfying a differential equation, or as the inverse of a logarithm (which in turn can be defined as an integral)).

Let's examine the definition of \(e\) as \(\exp(1)\), where \(\exp\) is the inverse function of \(\log\) with \(\log t=\int_0^t\frac1x\operatorname{d}x\). We have integrals, we have inverse functions, and then we have evaluation. This is not a first-order definition over the ordered field \(\RR\). It's barely even a second-order definition.

So even naively, we cannot really make any appeal to first-order definability over \(\Bbb R\) as an ordered field (if we could, then the algebraic reals would already include all the nameable numbers; whereas \(\pi\) and \(e\) are transcendental, they would be without names). Maybe we can say that a real number has a name, if there is some \(n\)-th order logic definition over \(\Bbb R\). And of course, that sounds pretty good. And we can prove, now, that there are only countably many definitions, so only countably many reals will have names, and therefore most reals are undefinable, unnameable, you name it.

Here we run into two problems very very quickly. The first is independence, using second-order logic we can write the statement "The continuum hypothesis holds", so if we are permitted to use second-order definitions, we can define the real which is \(1\) if the continuum hypothesis is true, and \(0\) is it is false. Is this real a nameable real? You know that it is either \(0\) or \(1\), but you have no practical way, sans adding more axioms to mathematics, to know which value it has.

Okay, so maybe that's cheating, maybe you want to be able to prove from \(\ZFC\) that a nameable number has a certain value. By this I mean, of course, that we should be able to decide with an algorithm whether or not a given rational number is smaller or larger than our nameable real. That sounds like a very cogent requirement from nameable reals. But again insufficient, simply because we can point at Chaitin's constant and say that it has a fairly definite definition. You could argue against this, and that is fine, but to say that Chaitin's constant is not a nameable number is essentially to say that only computable reals are nameable numbers. That is a valid approach to constructive mathematics, but it is not without difficulties. And since we want our mathematics to be simple and easy to use, computable analysis is not the road to take here.

The second problem, which is equally bad, is that \(n\)-th order logic definitions are simply not sufficient. It is not hard to come up with a real number which is not definable by any \(n\)-th order formula, but has a definite definition. Definite enough that we may consider it nameable. (And we should consider it nameable if we allow the "continuum hypothesis real" to be nameable!)

Set theory to the rescue? Okay, this is not really about set theory. This is about a foundational theory. Whichever it is that you like. If it allows some notion of real numbers which is "adequate", then it will usually come (or be bi-interpretable with) some mathematical world in which these real numbers live. So we can ask that a nameable number is a number which can be definable in such fixed world. And here comes the kick ass result, that it is possible that every real number is definable. Does it mean that the real numbers are countable? Yes, but not necessarily inside the model. Since the definitions now are not "inside" the mathematical world, but rather "outside" that world (they are in the foundational theory's meta-theory), we cannot quantify over the definitions and we cannot definably match a real with its definition.

So we get a universe where every real number is definable. And that is pretty amazing. Note that we've switched from the problematic nameable to the mathematical "definable". Because the notion of "nameable" is not really a mathematical notion. We like to think about numbers with names as numbers which come up organically, and naturally, from our lives or nature. But the truth is that we cannot know or not know what had, have, will or hadn't, haven't and won't come up and how these numbers are aligned with our perception.

And the major issue with names and definitions is that they live in the meta-theory. If you work with the real numbers, your meta-theory is "mathematics as you see it" or however you chose to formalize the notion of the real numbers and proofs and so on (let's say, for the sake of things and since it's my website, that \(\ZFC\) is your choice). So now \(\ZFC\) is your meta-theory. But once you allowed definitions not to come from some language about the real numbers, but to come from the entire wrath, might and power of your meta-theory, suddenly you find yourself appealing to its meta-theory.

If you've gone cross-eyed, don't worry. You're in good company. And what is my conclusion, then? That we should be remember that we formalize mathematics for a reason, and that some concepts like "nameable" are too ideal (in the Platonic sense of the word) to be given an explicit interpretation. Just like any definition of a chair is either circular (a chair is a chair), excessive (non-chair objects satisfy it), or insufficient (some chairs do not satisfy the definition).

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