The Universe of Discourse

Representing ordinal numbers in the computer and elsewhere
Lately I have been reading Andreas Abel's paper "A semantic analysis of structural recursion", because it was a referred to by David Turner's 2004 paper on total functional programming.

The Turner paper is a must-read. It's about functional programming in languages where every program is guaranteed to terminate. This is more useful than it sounds at first.

Turner's initial point is that the presence of ⊥ values in languages like Haskell spoils one's ability to reason from the program specification. His basic example is simple:

        loop :: Integer -> Integer
        loop x = 1 + loop x

0 = 1

Before you can use reasoning as simple and as familiar as subtracting an expression from both sides, you first have to prove that the value of the expression you're subtracting is not ⊥.

By banishing nonterminating functions, one also banishes ⊥ values, and familiar mathematical reasoning is rescued.

You also avoid a lot of confusing language design issues. The whole question of strictness vanishes, because strictness is solely a matter of what a function does when its argument is ⊥, and now there is no ⊥. Lazy evaluation and strict evaluation come to the same thing. You don't have to wonder whether the logical-or operator is strict in its first argument, or its second argument, or both, or neither, because it comes to the same thing regardless.

The drawback, of course, is that if you do this, your language is no longer Turing-complete. But that turns out to be less of a problem in practice than one would expect.

The paper was so interesting that I am following up several of its precursor papers, including Abel's paper, about which the Turner paper says "The problem of writing a decision procedure to recognise structural recursion in a typed lambda calculus with case-expressions and recursive, sum and product types is solved in the thesis of Andreas Abel." And indeed it is.

But none of that is what I was planning to discuss. Rather, Abel introduces a representation for ordinal numbers that I hadn't thought much about before.

I will work up to the ordinals via an intermediate example. Abel introduces a type Nat of natural numbers:

Nat = 1 ⊕ Nat

The ⊕ operator is the disjoint sum operator for types. The elements of the type S ⊕ T have one of two forms. They are either left(s) where s∈S or right(t) where t∈T. So 1⊕1 is a type with exactly two values: left(•) and right(•).

The values of Nat are therefore left(•), and right(n) for any element n of Nat. So left(•), right(left(•)), right(right(left(•))), and so on. One can get a more familiar notation by defining:

0	=	left(•)
Succ(n)	=	right(n)

And then one just considers 3 to be an abbreviation for Succ(Succ(Succ(0))) as usual. (In this explanation, I omitted some technical details about recursive types.)

So much for the natural numbers. Abel then defines a type of ordinal numbers, as:

Ord = (1 ⊕ Ord) ⊕ (Nat → Ord)

We can define abbreviations:

Zero	=	left(left(•))
Succ(n)	=	left(right(n))
Lim(f)	=	right(f)

So 0 = Zero, 1 = Succ(0), 2 = Succ(1), and so on. If we define a function id which maps Nat into Ord in the obvious way:

        id :: Nat → Ord
        id 0       = Zero
        id (n + 1) = Succ(id n)

        plusomega :: Nat → Ord
        plusomega 0       = Lim(id)
        plusomega (n + 1) = Succ(plusomega n)

        + :: Ord → Ord → Ord
        ord + Zero    = ord
        ord + Succ(n) = Succ(ord + n)
        ord + Lim(f)  = Lim(λx. ord + f(x))

Then this function multiplies a Nat by ω:

        timesomega :: Nat → Ord
        timesomega 0       = Zero
        timesomega (n + 1) = ω + (timesomega n)

But here's what puzzled me. The ordinals are really, really big. Much too big to be a set in most set theories. And even the countable ordinals are really, really big. We often think we have a handle on uncountable sets, because our canonical example is the real numbers, and real numbers are just decimal numbers, which seem simple enough. But the set of countable ordinals is full of weird monsters, enough to convince me that uncountable sets are much harder than most people suppose.

So when I saw that Abel wanted to define an arbitrary ordinals as a limit of a countable sequence of ordinals, I was puzzled. Can you really get every ordinal as the limit of a countable sequence of ordinals? What about Ω, the first uncountable ordinal?

Well, maybe. I can't think of any reason why not. But it still doesn't seem right. It is a very weird sequence, and one that you cannot write down. Because suppose you had a notation for all the ordinals that you would need. But because it is a notation, the set of things it can denote is countable, and so a fortiori the limit of all the ordinals that it can denote is a countable ordinal, not Ω.

And it's all very well to say that the sequence starts out (0, ω, 2ω, ω², ω^ω, ε₀, ε₁, ε_ε0, ...), or whatever, but the beginning of the sequence is totally unimportant; what is important is the end, and we have no way to write the end or to even comprehend what it looks like.

So my question to set theory experts: is every limit ordinal the least upper bound of some countable sequence of ordinals?

I hate uncountable sets, and I have a fantasy that in the mathematics of the 23rd Century, uncountable sets will be looked back upon as a philosophical confusion of earlier times, like Zeno's paradox, or the luminiferous aether.

[ Addendum 20081106: Not every limit ordinal is the least upper bound of some countable sequence of (countable) ordinals, and my guess that Ω is not was correct, but the proof is so simple that I was quite embarrassed to have missed it. More details here. ]

[ Addendum 20160716: In the 8 years since I wrote this article, the link to Turner's paper at Middlesex has expired. Fortunately, Miëtek Bak has taken it upon himself to create an archive containing this paper and a number of papers on related topics. Thank you, M. Bak! ]

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