Church integer

A representation of integers as functions invented by Alonzo Church, inventor of lambda-calculus.

The integer N is represented as a higher-order function which applies a given function N times to a given expression.

In the pure lambda-calculus there are no constants but numbers can be represented by Church integers.

A Haskell function to return a given Church integer could be written:

church n = c where c f x = if n == 0 then x else c’ f (f x) where c’ = church (n-1)

A function to turn a Church integer into an ordinary integer:

unchurch c = c (+1) 0

See also von Neumann integer.