Godel Numbering

[Note: this is a paste from another forum. Feel free to ignore-I don't mean to be spamming!]

I am trying to understand Godel numbering, and any help would be much appreciated. For fun, I was just trying to make some Godel numbers for functions (is this even allowable?) to try to investigate properties of these functions with arithmetical proofs on the Godel numbers. So I had a few questions:

1. I can use induction on Godel numbered statements, right?

2. Is it fine to number functions?

3. If it is fine, how do I represent operations arithmetically. For example, how could I arithmetize something like g(f(x))?

There was a book I saw recently in the bookstore about Godel's proof.  The thing is, I haven't read that one.  In fact I probably won't read it.  An FYI if you are interested in Godel's proof.  At this time I don't know any math professors that know alot about Godel's proof.  Though I could check with one that I know and ask some questions.  That said, I took an applied math course that had a brief topic of discussion along the lines of #3, though I'm not sure I have my notes handy.  The course was an applied math course in differential equations.  The instructor happened to talk about the sort of thing alluded to in the third point. 

By the third point, did you mean something that might be in Calculus level or more advanced describing it mathematically.  I'll try to look this up, though admitedly I'm not an expert on Godel's proof.

About the third question, I was just wondering what a useful translation scheme would be; from my superficial understanding, the point would be to translate certain statements into other statements that one can find proofs of in number theory. For example, if I was wondering whether or not some statement in a formal system was a theorem, then I could translate the statement and the system's rules into number theoretic statements in order to use proofs from number theory that when retranslated, would give me proofs about the original statement. The main question, then, is a useful translation of rules (like allowing for a composition of functions) that could be easier to work with when translated.

Godel chess engine for Droidfish