Fully general implies recursively axiomatizable?

I keep reading things in the history of phil math which seem to assume that:
'Theory t is fully general' implies 'Theory t is logical' implies 'Theory t recursively axiomatizable'.

e.g. Frege thought that math might be a matter of logic, i.e. fully general principles of reasoning. But then we discovered that his particular axiomatization didn't work, and we learned from godel that no recursively enumerable axiomatization could capture all mathematical truths. Hence, we learned that math is not a matter of pure logic, and that it is not fully general, but rather contains subject-matter specific truths.

But why are we assuming that being "fully general" implies being recursively axiomatizable? Can anyone else see an argument for this claim?

[I also feel like there's a tendency to assume 'fully general' implies 'logical' implies 'epistemically unproblematic'. But, e.g. second order logic would seem to be fully general while still being epistemically problematic. So, again, I'd like to know what argument there might be for this.

Indeed, its not clear to me what, if any, entailment relations between:
1. is logical 2. is recursively axiomatizable 3. is epistemically unproblematic 4. must be accepted by any thinker 5. is fully general 6. is true in virtue of meaning 7. can be known merely be reflecting on meaning 8. is epistemically unproblematic.]