We have some understanding of natural numbers. We have PA theory and we believe that $\mathbb N$ is one of the PA models. But PA can't prove some statements about $\mathbb N$ even though they are true in this specific PA model. And in fact there are no any finite system of axioms which would be able to completely describe $\mathbb N$. All we can do is to specify new axioms to create more complete formal description of $\mathbb N$.
But now here is a question: do we really have complete understanding of natural numbers? Let me put it in other words. Suppose there is a program which asks you a list of questions about $\mathbb N$. And let's suppose that when you answer a question from the list you actually give more precise formal description of $\mathbb N$. And after you answer some questions from the list the program will give you some new axioms based on your answers. The question is whether we can actually answer all questions from this list based only on our "natural" understanding of natural numbers.
There is a trivial example that illustrates that the answer is negative. The example is the question about consistency of theory consisting of all true statements about $\mathbb N$. We can't answer it based only on our "natural" understanding of natural numbers. But let's maybe ignore this question. Will we be able to answer any other question?
If the answer is negative then do I understand correctly that actually we do not understand what natural numbers are? And when we create more strong axiomatic system then we actually come up with more complete understanding of natural numbers?
