How Many Numbers Can a Lambda-Term Contain?

Abstract

It is well known, that simply-typed λ-terms can be used to represent numbers, as well as some other data types. We prove, however, that in a λ-term of a fixed type we can store only a fixed number of natural numbers, in such a way that they can be extracted using λ-terms. More precisely, while representing k numbers in a closed λ-term of some type we only require that there are k closed λ-terms M ₁,…,M _k such that M _i takes as argument the λ-term representing the k-tuple, and returns the i-th number in the tuple (we do not require that, using λ-calculus, one can construct the representation of the k-tuple out of the k numbers in the tuple). Moreover, the same result holds when we allow that the numbers can be extracted approximately, up to some error (even when we only want to know whether a set is bounded or not).