Answer

Thank you. I read it, and it was interesting.
I don’t see any innovation here with principled significance. It has long been known (since 1970) that Diophantine equations are undecidable over the integers. So what is the principled significance of the fact that they are also undecidable over other number rings? The fact that our ability to prove or decide is limited was already known beforehand. So philosophically, what difference does it make if there is one more undecidable problem?
The superlatives at the beginning of the article (about important findings that touch on the limits of our knowledge) are characteristic of popular science writing, where they always try to turn every finding into something revolutionary and philosophical.
By the way, the decision problem and Turing’s halting theorem apply only to machines that are equivalent to a Turing machine. That does not mean that more sophisticated machines (quantum computation, for example) will not be able to decide everything. Just as I noted in the lecture about systems stronger than arithmetic, to which Gödel’s theorem does not apply.
I am not sufficiently expert in computability theory, and I do not know what theorems have been proved regarding quantum computation or how much stronger it is than a Turing machine. But my intuition tells me that because of the analogy between the halting theorem and Gödel’s theorem, if there are axiomatic systems in mathematics to which Gödel’s theorem does not apply, then there are probably also computational machines to which the halting theorem does not apply. I don’t know whether these are quantum computers or something even stronger. But there ought to be such machines.
Goodbye,