I didn't confuse completeness in the sense of the Completeness Theorem with the sense in the Incompleteness Theorem. I made two separate points. One point was

I didn't confuse completeness in the sense of the Completeness Theorem with the sense in the Incompleteness Theorem. I made two separate points. One point was to state what the Incompleteness Theorem means and the other was to show that what you wrote was wrong. You made the claim: "It says that in any sufficiently complex axiom system there will be statements that it can neither prove nor refute,... This is clearly false unless you restrict yourself to recursively enumerable axiomatic systems. My proof of this was to take all true statements of the model of the integers under the second order system. That forms an axiomatic system in which all true statements are provable. That is all I have really said in this matter. The recursively enumerable criterion is important because of how all this came about historically. Humans - Hilbert, Russell, etc. - had a perception of what the integers are. Their goal was to encapsulate mathematical truth 'algorithmically' (in a computable way) and Godel showed this wasn't possible. All true statements of the first order axioms are true in the model of the second order axioms. The reverse isn't true. The dream was crushed. They didn't know before Godel that the natural numbers we all grew up with couldn't be modeled properly with a recursively enumerable set of axioms but that's what they wanted and were looking for. EDIT: I conflated the terms 'refute' and 'disprove'. Sorry about that. There is a difference between proving ~A and finding a counterexample to A in some model. I read 'refute' as 'disprove'.