My representation is countable, but not because of the fact it uses the algebraic term 'r * r' in its definition of sqrt(2). My representation is based on dede

My representation is countable, but not because of the fact it uses the algebraic term 'r * r' in its definition of sqrt(2). My representation is based on dedekind cuts, implemented in Haskell, which is based on the lambda calculus. There are countably many lamdda terms. Thus, I can only represent countably many real numbers. My representation can store all the computable numbers, but there are computable numbers that are not algebraic. For example, e is not algebraic, but it is computable. Here is a representation of e factorials :: Rational factorials = 1:zipWith (*) [1..] factorials eApproxes :: [Rational] eApproxes = scanl (+) 0 (map recip factorials) e :: Rational e = Rational (\r -> any (r <) eApproxes)