Ariithmetic is complex because it can not be captured in a small set of axioms.

What is this "it"? There are some who claim that when we think about arithmetic, we are thinking about a specific model of the usual axioms for arithmetic, which appears to be your view here. Every statement of arithmetic is either true or false in that model. But what reason is there to make this claim? We cannot directly intuit the truth of arithmetical statements, or mathematicians would not have to spend so much effort on proving theorems. We may observe that we have a belief that we are indeed thinking about a definite model of the axioms, but why should we believe that belief?

To say that we intuit a thing is no more than to say we believe it but do not know why.