For those who are getting confused about the S and K combinators, another definition is implied by the notion of a 'partial combinatory algebra' [1]. It is a set A together with a partial map A x A -> A, viewed as 'application'. You then require the S and K combinators to exist somewhere in A (they are non-unique) and you can prove things like the recursion theorem.

[1] https://ncatlab.org/nlab/show/partial+combinatory+algebra