The incompleteness theorem says that a consistent axiomatic formal system satisfying some conditions cannot be complete, so the universe as a formal system (supposed consistent, complete, expressive enough, …) cannot be axiomatized.

It can also be axiomisable but inconsistent. In principle, that is, but as said you'd annoy a lot of physicists.

What do you mean external?

As in the previously mentioned summation of the results of theoretical hypercomputation: "If uncomputable inputs are permitted, then uncomputable outputs can be produced". Those oracles would be the input.

The possibility of using physical phenomena as oracles for solving classically uncomputable problems in the real world is an open question.

If they exist, then they can be used. We do that all the time in the sense that we're pretending they exist, it's useful to e.g. prove that an algorithm is optimal: We compare an implementable algorithm it with one that can e.g. see the future, can magically make all the right choices, etc. But they don't exist.

If you think this is logically as impossible as a four sided triangle you should give sources for this claim

I already pointed you to an easy-going explanation of the proof by diagonalization. I'm not going to sit here and walk you through your homework. In fact I have given up explaining it to you because you're not putting in the work, hence why I resorted to an analogy, the four-sided triangle.

Some undiscovered physical phenomenon might make this possible… who knows.

Are all thinkable phenomena possible? Can there be four-sided triangles?

The four sided triangle is logically impossible, but a hypercomputer is logically possible.

That is an assertion without substantiation, and for what it's worth you're contradicting the lot of Computer Science. A hypercomputer is a more involved, not as intuitive, four-sided triangle.

If you think that it's logically possible, go back to that proof I pointed you to. I will not do so again.