Well, let's have the various ways for representing recursive functions. The crucial ingredient is the fix point operator, which allows to define functions by recursions by doing that as a fix point of a suitable operator.
The fix point, you see, has a very simple syntax is enough to formulate such a complicated concept.
There is another way of encoding the fix point recursive function and this is by introducing suitable constance called the recursors, regulated by special reduction rules, called d-reduction rules.
So suppose you introduce the special constance of natural numbers, as a 0 and successor, then you can define a new constant a recursor, which, given two parameters a and b, allows you to define a function on by recursion.