dwohnitmok, I think you make a good point about Completeness missing out on the action in OP's post, which is nevertheless a good post and better than popular treatments. In fact, I remember thinking in class, "What, there is Completeness theorems as well as Incompleteness?" This is due to a similar phenomenon as we see with quantum mechanics in popular writing, where mostly the sensational, which aren't really sensational, is written.
In any case, my contribution here is something else: The most interesting part of the natural numbers object and its role in incomplete logical systems is to me the more humble notion of... unique prime factorisation.
If that had failed, so too would the Godel numbering system. I am not a number theorist, but this to me would be an interesting spin off from OP's post. The divisibility lattice is a good starting point, but apart from such basic constructions, the actual proof of unique prime factorisation to me is more historic than informative, and steps such as Bezout's Identity are just as "mythical" to me as Godel's Incompleteness Theorem, if we are using bad word choices. Conversely, numbers are just permutations on prime generators, but crucially, usually we start with addition before we start with multiplication.
In any case, my contribution here is something else: The most interesting part of the natural numbers object and its role in incomplete logical systems is to me the more humble notion of... unique prime factorisation.
If that had failed, so too would the Godel numbering system. I am not a number theorist, but this to me would be an interesting spin off from OP's post. The divisibility lattice is a good starting point, but apart from such basic constructions, the actual proof of unique prime factorisation to me is more historic than informative, and steps such as Bezout's Identity are just as "mythical" to me as Godel's Incompleteness Theorem, if we are using bad word choices. Conversely, numbers are just permutations on prime generators, but crucially, usually we start with addition before we start with multiplication.