We are all familiar with Cantor’s diagonal argument that proves there exist infinite sets which are “larger” than the set of natural numbers. In this post I will show that we can express this argument in the form of a program, thus showing that there are countable sets which are “computably uncountable”.
I begin with the program itself:
type Cantor = Nat -> Bool
diagonal :: (Nat -> Cantor) -> Cantor
diagonal cs n = not (cs n n)
Cantor is “the cantor space”, the type of infinite sequences of booleans. We will call such an infinite sequence “a Cantor“. There are clearly infinitely many Cantors; e.g. take the range of this function which gives False at every position except the one specified:
unit :: Nat -> Cantor
unit m n = m == n
diagonal is (Georg) Cantor’s diagonal argument written as a program — it takes an alleged sequence of all Cantors, and returns a Cantor which does not occur in the sequence, by construction. This function shows by contradiction that we cannot put Cantors in 1 to 1 correspondence with naturals, and thus that there are more Cantors than there are naturals.
So how many Cantors are there? Since Nat -> Bool is a Haskell type — the type of computable functions from Nat to Bool — Cantors must be representable by programs. We can encode programs as numbers by treating their source code as base-128 numbers. Hence, there are no more Cantors than naturals, and so Cantors can be put into 1 to 1 correspondence with naturals.
Wait — what? There are more Cantors than Nats, but they both have the same size? Something is wrong. Indeed, in the process of this argument we have asserted both
- “We cannot put Cantors in 1 to 1 correspondence with naturals”
- “Cantors can be put into 1 to 1 correspondence with naturals”
We clearly can’t have both.
The erroneous statement is (2). It is undecidable whether a given program represents a Cantor. If the nth Cantor is ⊥ at n, then diagonal will fail: diagonal cs n = not (cs n n) = not ⊥ = ⊥. Because ⊥ is a fixed point of not, diagonal cannot return an element different from the one it was given. Thus for diagonal to work, we must require that Cantors be fully-defined — no infinite loops!
With this requirement, we can no longer put Cantors in 1 to 1 correspondence with the naturals, because we would have to solve the halting problem. It is not enough that the type of the term is a Cantor, it now must be fully defined for all inputs, and determining that given arbitrary source code is an undecidable problem.
The erroneous statement is (1). Cantors are computable functions, so as we have argued, they have the same cardinality as the naturals. There are no more programs than numbers, so by the definition of equal cardinality we can put them in 1 to 1 correspondence with a function.
The problem with (1) occurs because diagonal takes as its first argument not an arbitrary sequence of Cantors, but a computable sequence of Cantors. If cs is not computable, then neither is diagonal cs (for we no longer have cs‘s source code with which to construct it), and Cantors are defined to be computable sequences. So diagonal fails to contradict our bijection.
The erroneous statement is (2). Section II claims to put Cantors and naturals in 1 to 1 correspondence, but it is lying. Suppose Section II is formulated with respect to some axiom system A. If it were “telling the truth”, we would expect there to be some term f in the language of A such that for every fully defined Cantor program c, there is some natural number n such that we have (i.e. it is a theorem of A that f(1 + 1 + … + 1) = (source code of c)).
Let’s suppose we have written down the axioms of A into a Haskell program, and we have a (partial) function proofSearch :: Nat -> Cantor, which, given a number n, searches for theorems of the form and compiles and returns the first such c it finds. In the case that there is no such statement, it just runs forever; similarly for the case that c fails to compile. Although cumbersome to write, I’m sure we agree that this is possible to write. If section II is not lying, then we expect that for every natural n, proofSearch n does in fact return a valid Cantor.
Now, let us return to familiar lands with a new program:
evidence :: Cantor
evidence = diagonal proofSearch
Oh my! If section II is the truth, then proofSearch is a total, computable function of type Nat -> Cantor, which we can pass to diagonal to find a Cantor that it missed! So it must have been lying, either (1) about its function f finding every possible Cantor or (2) about it actually possessing such a function (i.e. it “proved” that there is such a function, but it couldn’t actually represent it). In either case, it did not actually create a 1 to 1 correspondence between the naturals and Cantors.
Left as an exercise for the reader.
Which one is it really?