I've been getting extremely conflicting answers about this. Please help me. A summary of the different viewpoints:
It seems like most people are taking the Platonic viewpoint : that natural numbers are taken for granted and statements about natural numbers are true or false in the absolute sense. They are false if a counter-example exists. And that we do not require any axiomatic system to make precise the notions of "natural numbers", "there exists", "counter-example", etc. All of it comes from human intuition about natural numbers
The second viewpiont is we really have two axiomatic systems: A strong one like $ZFC$ and a weaker one like Peano arithmetic. The stronger model is taken to define natural numbers. Anything that's provable about the naturals in the stronger model is referred to as the "truths" about naturals. So the weaker axiomatic system like Peano is incomplete in the sense that it cannot derive all the truths of the stronger model. Also there is the non-standard model of arithmetic. So, the reason Peano axioms are incomplete is really because there are no absolute truths about natural numbers. The truths depend on our definition of natural numbers according to the stronger theory. Does the non-standard model of arithmetic disprove the Platonic viewpoint?
What I find weird about the second viewpoint is that it makes Godel's incompleteness theorem seem trivial. It means the thoeorem just says that certain statements about natural numbers may be independent from Peano arithmetic in the same sense the continuum hypothesis is independent from ZFC. Those statements are true in some models and false in others.
If the second viewpoint is correct, then does it mean that human intuition about natural numbers is actually ambiguous? Humans feel like we just know, without any reference to any axiomatic system, what it means for statements about natural numbers to be true or false : it's just false if you can find a counter-example. Is this intuition wrong and the statements are actually ambgious (due to e.g. the non standard model of arithmetic)?
OR Is the intuition actually correct and it's just that any axiomatic system (even ZFC) is insufficient to model the the natural numbers that unambiguously exist in human intuition. If yes, on what basis can we claim this?
And since Godel's incompleteness theorem is related to the halting problem, how do the two viewpoints relate to the halting problem? It seems like the halting problem is making an absolute statement about the limits of computability using Turing machines (e.g. the Busy Beaver function is non-computable in the absolute sense).