My course is about computability theory, and I'm having troubles with one of the main concepts. This might be a really newb question, but I've been struggling with understanding it's significance (and I was stupid enough to not ask my teacher while in class).
Can someone give me an example in terms of something practical to why numbering computable functions is helpful?
I'm a computer science student, so maybe an example from that field might help me understand.
EDIT: Okay, so with the assumption then that the halting problem is the main reason we number computable functions.
In that case, numbering the functions should help us always prove if a function will halt or not. So, say we take the gödel numbered program $P_{2057}$, which stands for the program S(1), S(1), J(2,1,1). This represents the function x+2, over the domain of $\mathbb{N}$.
How does knowing the number 2057 help us decide if this program will halt or not?
EDIT2: I tried reading the papers that @sav linked to me, although I find them relatively difficult to understand, as I'm not that good with theoretical things yet.
Our course book is Computability, An introduction to recursive function theory by Nigel Cutland. I've found online many people praising this book for being clear and easy to understand, but for me that hasn't always been the case. The teachers explanations don't always make sense either, and sometimes it feels like the book is the teacher, not her(which is one of the big reasons I finally decided to come here for help).
Therefore, another question I have is, the book claims the numbering of functions to be of utmost importance in both the s-m-n theorem, and universal functions. After reading through those parts multiple times, I still don't understand their significance, and yet they are constantly referenced in later parts. Could someone maybe try and explain this connection in another way?