What is the significance of algebraic datatypes with zero constructors?

Question

This passage, which unfortunately lacks references, about the development of ADTs in Haskell, from A History of Haskell: Being Lazy With Class, section 5.1:

Answer 1

Among corresponding to logical false (as stated in another answer), they are often used to create additional type constraints in combination with GADTs. For example:

{-# LANGUAGE GADTs #-}
{-# LANGUAGE EmptyDataDecls #-}

import Data.List (groupBy)

data Zero
data Succ n

data Vec n a where
    Nil  ::                 Vec Zero a
    Cons :: a -> Vec n a -> Vec (Succ n) a

vhead :: Vec (Succ n) a -> a
vhead (Cons v _) = v

vtail :: Vec (Succ n) a -> Vec n a
vtail (Cons _ v) = v

Here we have two such data types with no constructor. Their meaning here is just to represent natural numbers: Zero, Succ Zero, Succ (Succ Zero) etc. They are used as phantom types in Vec data type so that we can encode the length of a vector in its type. Then, we can write type-safe functions such as vhead/vtail that can be applied only to non-empty vectors.

See also [Haskell] Fixed-length vectors in Haskell, Part 1: Using GADTs where the example is elaborated in detail.