Godel's Incompleteness Theorems

Godel Background
The First Incompleteness Theorem
In simplest terms, Godel's First Incompleteness Theorem states that: for systems of sufficient strength (such as arithmetic with multiplication) the system is either inconsistent or incomplete.

Godel proves this by proving that it is possible to construct (in arithmetic with multiplication) a sentence that we can interpret as meaning, This sentence is not provable. We call this now a Godel Sentence. Note: all he has to do is prove this can be said in arithmetic, and as a consequence arithmetic must be either inconsistent or incomplete. Can you see why?

The proof is difficult, but we can consider it in the abstract, and also consider some simplified and analogous proofs.

Smullyan's helpful analogies
We have a theory with:
Let's remind ourselves of some terminology. Here is a rough description of our main concepts (they need to be precisely defined relative to a particular logical system--but don't worry about that detail): I'd like for you to be able to take away the following: