I'm trying to get a better grasp of the relationship between these concepts so I have a few questions about them.
Does a consistent theory have to be (or have to not be) axiomatizable?
Does an inconsistent theory need to be axiomatizable? (Is the answer yes because all inconsistent theories are the same in that anything is provable?)
Does an axiomatizable have to be decidable?
And are decidable theories always complete/what's their relation?
A specific example for some of these is robinson arithmetic which I know is axiomatized and incomplete. Is it decidable? Could it be extended to be decidable or complete? Would this be an example of an axiomatizable theory that's not decidable?
Sorry if this is scattered, there are a lot of relationships here I'm not sure how to connect.