Thread: Godel Numbering

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.

Originally Posted by charmedquark

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.

Hopefully you know what you're talking about but us, the rest of poor mortals, most probably

have no clue, and unless you tell us (What are you "desired expressions", what are

"strings"...??), there's a very poor chance someone will be able to help you.

Tonio

For example the whole point is to have a tool for certain statements to talk about themselves...for example, I could arbitrarily have:


Map
x0<----->0
0<------>1
*<------>2
(<------>3
+<----->4
1<------>5
^<----->6
=<----->7
2<----->8
etc.


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?

Originally Posted by charmedquark

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.

Assuming you are talking about Goedel numbering, as in your title, then, generally speaking, you don't. It is not the point of Goedel numbering to "derive theorems".

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 thread belongs in the Logic forum.

Godel numbering is essential in proving Godel incompleteness theorems (at least via diagonalization).

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.