>to think that we should have all these simple axioms, and then out of left field accept the continuum hypothesis as an axiom is a bit weird don't you think?
It's like the parallel postulate - famously much more awkward and complicated than the other axioms of geometry, and you can do most of geometry the same with or without it. But even so, most of us choose to use standard euclidean geometry and accept the parallel postulate.
>also weird is that we can build up all this mathematical machinery from the ZF axioms, but still can't answer some seemingly innocuous questions.
I think of it like this: "the language of mathematics is expressive enough that we can formulate nonsense questions". If you look at it that way the incompleteness theorem isn't terribly surprising; in natural language any five year old can ask unanswerable questions.
It's like the parallel postulate - famously much more awkward and complicated than the other axioms of geometry, and you can do most of geometry the same with or without it. But even so, most of us choose to use standard euclidean geometry and accept the parallel postulate.
>also weird is that we can build up all this mathematical machinery from the ZF axioms, but still can't answer some seemingly innocuous questions.
I think of it like this: "the language of mathematics is expressive enough that we can formulate nonsense questions". If you look at it that way the incompleteness theorem isn't terribly surprising; in natural language any five year old can ask unanswerable questions.