Constructability, Uncountability, and ω-Haskell

My last post about the uncountable ordinals ended up on the programming subreddit, where someone who I think is Lennart Augustsson said:

Wow, I didn't realize you could do this. Now I have to wrap my brain around in what sense you can represent uncountable ordinals in Haskell since there are only a countable number of Haskell programs. It must be something like constructive reals...

I think he's exactly right, and I think it has something to do with V=L, models, and how many functions are definable in Haskell. I'm going out of my comfort zone on this, so please correct me if I make any incorrect or nonsensical statements. I'll use as my example the powerset of the natural numbers, which is certainly uncountable, and already definable in Haskell with

data Nat = Z | S Nat
type Powerset t = t -> Bool

From the viewpoint of ZFC, there are an uncountable number of ZFC functions from the natural numbers to the booleans. From the viewpoint of Haskell, there are an uncountable number of Haskell functions from the natural numbers to the booleans. But, from the world of ZFC, the number of Haskell functions from the natural numbers to the booleans is countable: we can simply order them by their definitions lexicographically. Since every function definable in Haskell has a finite definition and the cardinality of a countable union of finite sets is countable, the function definitions of type Nat -> Bool are countable.

|{f : f <- (N -> 2)_ZFC}| >_ZFC |N|
|{f : f <- (N -> 2)_Haskell}| >_Haskell |N|
|{f : f <- (N -> 2)_Haskell}| =_ZFC Σ_{n <- N} |all n-bit functions from N -> Bool| = |N|

From the viewpoint of Haskell, though, this can't be proven; functions are opaque, so we have no way to order or count them. This might not be true if we use some of the unsafe features of Haskell that would allow us to look at the actual assembly corresponding to a function.

We could, I suppose, imagine a programming language which was not limited to finite-length function definitions. ω-Haskell would not only have types with seemingly uncountable numbers of inhabitants, but each inhabitant would also be definable. Yuk!

Countable Ordinals in Haskell

In Haskell: What happens when you divide infinity by 2?, Eric Kidd asks:

...
2. Is it possible to represent any other (more interesting and/or more correct) definitions of infinity in Haskell?
3. What’s the best way to think about the infinite ordinals?

The code below is my answer to those questions for countable ordinals only. I'm not sure if it makes sense to talk about representing uncoutable sets on the kind of computers we have now: We can talk about infinite countable sets as computable limits of finite objects, but what would an uncountable set look like? They can't be objects of Nat -> Bool, since every object of this type is represented in some finite number of bits. It seems like we are stuck in a universe where Goedel's V=L is true.

{-# OPTIONS_GHC -fglasgow-exts #-}

{- See:
  http://en.wikipedia.org/wiki/Ordinal_arithmetic
  http://en.wikipedia.org/wiki/Large_countable_ordinals

  To add later:
  http://citeseer.ist.psu.edu/487179.html
-}


module Countable where

data Finite = Z
           | S !Finite

{- A countable ordinal is 0, a successor, or a limit of a countable
  sequence. This definition is weak, as the countable sequence should
  be strictly increasing, but that's hard to express. One way to do so
  would be to interpret (Limit f) as the limit of
  (sum i = 0 to n) (1 + f i). Note: this addition is not commutative.
  This would complicate things, however, so I'll leave it for a later
  extension.
-}
data Ordinal = Zero
            | Succ !Ordinal
            | Limit (Finite -> Ordinal) deriving (Show)

{- I believe that there is no computable Eq or Ord instance for Ordinals.
  Comments?
 
  We make an approximation here by taking the nth value of the limit.
  This calculation is very slow, and, of course, will equate a lot of
  ordinals that are not actually equal.
-}

nth :: Ordinal -> Finite -> Integer
nth Zero _ = 0
nth (Succ x) n = 1+(nth x n)
nth (Limit f) n = nth (f n) n

instance Show (Finite -> Ordinal) where
   show f = show [f Z,
                  f (S Z),
                  f (S (S Z)),
                  f (S (S (S Z)))]

instance Eq Ordinal where
   x == y = (show x) == (show y)

{- The basic ordinal arithmetic. Addition and multiplication are not
  commutative.
-}
addOrdinal :: Ordinal -> Ordinal -> Ordinal
addOrdinal x Zero = x
addOrdinal x (Succ y) = Succ (addOrdinal x y)
addOrdinal x (Limit y) = Limit (\z -> addOrdinal x (y z))

multiplyOrdinal :: Ordinal -> Ordinal -> Ordinal
multiplyOrdinal x Zero = Zero
multiplyOrdinal x (Succ y) = addOrdinal (multiplyOrdinal x y) x
multiplyOrdinal x (Limit y) = Limit (\z -> multiplyOrdinal x (y z))

expOrdinal :: Ordinal -> Ordinal -> Ordinal
expOrdinal x Zero = Succ Zero
expOrdinal x (Succ y) = multiplyOrdinal (expOrdinal x y) x
expOrdinal x (Limit y) = Limit (\z -> expOrdinal x (y z))

-- The first infinite ordinal
finite :: Finite -> Ordinal
finite Z = Zero
finite (S x) = Succ (finite x)

omega = Limit finite

-- Apply a f some number of times, starting at z
apply Zero f z = z
apply (Succ n) f z = f (apply n f z)
apply (Limit g) f z = Limit (\x -> apply (g x) f z)

-- With apply, we have more succinct definitions of arithmetic
addOrdinal' x y = apply y Succ x
multiplyOrdinal' x y = apply y (flip addOrdinal x) Zero
expOrdinal' x y = apply y (flip multiplyOrdinal x) (Succ Zero)

-- Enumerates the fixed points of a function.
fix Zero f = apply omega f Zero
fix (Succ n) f = apply omega f (Succ (fix n f))
fix (Limit g) f = Limit (\x -> fix (g x) f)

-- Some larger ordinals:
epsilon n = fix n (expOrdinal omega)

-- The Veblen heirarchy:
veblen Zero b = expOrdinal omega b
veblen (Succ a) b = fix b (veblen a)
veblen (Limit a) b = Limit (\x -> veblen (a x) b)

-- Feferman-Schutte ordinals
gamma n = fix n (flip veblen Zero)