There is no such thing as a "complete theory of arithmetic" - see Godel's incompleteness theorems.
The above statement isn't true. What Godel's theorem says is that any effectively generatable theory which is strong enough to model PA is either incomplete or inconsistent. Effectively generatable for our purposes can be taken to mean that valid proofs in the system are enumerable and that there's a primitive recursive procedure to determine whether a given proof is valid.
If one removes the effectively generatable condition then one can easily produce a complete, consistent system. Consider for example the following system: Consider some lexicographic ordering of well-formed statements in the language of PA, and let S(i) be the ith statement in this list. Let P(0) be PA, and generate P(n) by asking whether S(n) or ~S(n) is a theorem in P(n-1). If so, then set P(n) = P(n-1). If not, then let P(n) be P(n-1) with the additional axiom of S(n). Now, consider P(infinity) defined as the union of all P(n). This theory is complete and consistent but it is very much not effectively generatable since we can't even recursively enumerate the system's axioms.
It might help for you to take a model theory or logic course since there are a lot of subtle issues here that you seem to be missing.
You know that automated proof verifiers exist, right? And also that programs can know their own source code? Well, here's a puzzle for you:
Consider a program A that knows its own source code. The algorithm of A is as follows: generate and check all possible proofs up to some huge size (3^^^^3). If A finds a proof that A returns 1, it returns 1. If the search runs to the end and fails, it returns 0. What will this program actually return?
Wait, that was the easy version. Here's a harder puzzle:
Consider programs A and B that both know their own, and each other's, source code. The algorithm of A is as follows: generate and check all proofs up to size 3^^^^3. If A finds a proof that A returns the same value as B, it returns 1. If the search fails, it returns 0. The algorithm of B is similar, but possibly with a different proof system and different length limit. What will A and B return?
This second puzzle is a formalization of a Prisoner's Dilemma strategy proposed by Eliezer: "I cooperate if and only I expect you to cooperate if and only if I cooperate". So far we only knew how to make this strategy work by "reasoning from symmetry", also known as quining. But programs A and B can be very different - a human-created AI versus an extraterrestrial crystalloid AI. Will they cooperate?
I may have a tentative proof that the answer to the first problem is 1, and that in the second problem they will cooperate. But: a) it requires you to understand some logic (the diagonal lemma and Löb's Theorem), b) I'm not sure it's correct because I've only studied the subject for the last four days, c) this margin is too small to contain it. So I put it up here. I submit this post with the hope that, even though the proof is probably wrong or incomplete, the ideas may still be useful to the community, and someone else will discover the correct answers.
Edit: by request from Vladimir Nesov, I reposted the proofs to our wiki under my user page. Many thanks to all those who took the time to parse and check them.