Of course, it is very simple to make an arbitrary map for my desired expressions into numerical strings, but how do I actually do something useful with the numerical strings? That is, what theorems could I derive from these numbers? Any examples would be appreciated.
For example the whole point is to have a tool for certain statements to talk about themselves...for example, I could arbitrarily have:
So that I can write 1+1=2 as 5,4,5,7,8-->54,578
and now, what useful things can I prove in arithmetic with a number like 54,578 that lets me implicitely make theorems about my other statements?
Are you sure? What is the purpose, then? I thought Godel wanted a way for number theory to talk about itself, and I would assume that just listing strings of arbitrary numbers wouldn't do much good to that end!
Then please tell me what I can do with this.
This book may not answer all of your questions, but it is not a bad place to start.
Gödel, Escher, Bach - Wikipedia, the free encyclopedia
It can be found in most libraries.