Church integer

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A Church integer is a representation of natural numbers as functions, invented by Alonzo Church as part of his lambda calculus. The natural number n is represented as a higher-order function, which takes a function f as argument and returns the n-fold composition f o f o ... o f as result.

For example, in Haskell, a function that returns a particular Church integer might be

church 0 = \\ f x -> x
church n = c
  where
    c f x = c' f (f x)
      where
        c' = church (n - 1) 

The transformation from a Church integer to an integer might be

unchurch n = n (+1) 0

Thus the (+1) function would be applied to an initial value of 0 n times, yielding the ordinary integer n.

See lambda calculus for another expression of the same idea.

See also: Gödel number.

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An earlier version of the above article was posted on PlanetMath. This article is open-content.

There are some interactive examples of Church integers here.