Godel's Incompleteness Theorem So I have to give a talk on Godel's Incompleteness Theorem in which I have to give a brief proof of Godel's Incompleteness Theorem in a non-techincal simple English way. The problem is I am not really too sure on how to take such a technical concept and make it non-technical while keeping the essence of the proof.
This is how I am stating the theorem:

Each computably axiomatizable L-theory extending Peano Axioms is incomplete.

Any help will be appreciated. (If you want me to provide any more info, like the technical proof I have let me know)
 A: For a non-technical talk, I would stick to this:
"There is no complete axiomatization of arithmetic"
That is, there is no finite set of axioms for arithmetic such that arithmetical truths can be derived from them.
(Godel showed there is no such recursive set, but for simplicity sake I would just stick to finite)
And for a proof sketch:
Assume you have a finite and sound (meaning that only arithmetical truths can be derived from it) set of axioms $A$ that is 'at least as strong as the Peano axioms'.
Then, show the basic idea of arithmetization, i.e. a way of assigning numbers to logical symbols, expressions, and derivations, so that statements about arithmetic can be treated as statements about logical statements and proofs. Just show some very basic examples of how such a coding scheme could work, and then just wave your hands and say that we can create a formula that effectively says 'the statement with Godel number $x$ is not provable from A', where the fact that we can have such a formula depends on $A$ being 'at least as strong as PA' and being finite.
Next (and again without any further proof) mention the Diagonal Lemma through which Godel established that for every formula there is a sentence that is true if and only if that formula is true for the godel number of that sentence. Thus, in particular, there will be a Sentence G (the Godel sentence) that ends up saying "I am not provable from A"
And now do the grand finale! If $G$ is false, then G is provable from $A$, but given that $A$ is sound, that means $G$ is true. Ok, so G cannot be false, so it is true. And so $G$ is indeed not provable from $A$. So, there is something that is true, but not provable from $A$, and thus $A$ is not complete.
Finally, you should probably mention that this works for any finite $A$. Thus, if someone says "So why don't we just add $G$ to $A$", then you say "Well, that means we have axiom set $A'$, and $A'$ has its own Godel sentence $G'$ that is true but cannot be proven from $A'$.
