# Equivalence between $\lambda$-calculus and recursive functions

I have two related questions:

1. Do you know of any good reference to the recursive function / lambda-calculus equivalence in terms of computability, including proofs?
2. Do you know of any reference to subclasses of computable layers in lambda-calculus, like you have in recursion theory? It looks like you can't "dumb down" lambda-calculus but I'm not sure about that. And if we can't "dumb it down" to less computable classes, I'd love some reference to why this is so.
• Although the questions are somewhat related, it might be better to split them. Sep 10, 2021 at 17:08
• For 1., see e.g. J. Hindley & J. Seldin (2008), Lambda Calculus and Combinators: An introduction, chapter 4. Sep 10, 2021 at 18:13