# Godel Numbering

• Feb 22nd 2011, 07:55 AM
charmedquark
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.
• Feb 22nd 2011, 09:01 AM
tonio
Quote:

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
• Feb 22nd 2011, 10:13 AM
charmedquark
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?
• Feb 24th 2011, 07:00 AM
HallsofIvy
Quote:

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".
• Feb 24th 2011, 11:19 AM
charmedquark
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.
• Feb 24th 2011, 11:39 AM
emakarov
This thread belongs in the Logic forum.

Godel numbering is essential in proving Godel incompleteness theorems (at least via diagonalization).
• Feb 24th 2011, 12:26 PM
TheEmptySet
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.