>>14179246His incompleteness theorems proved that a system like Principia can never be consistent and complete at the same time. Either it will be complete but self-contradictory, or consistent (no contradictions) but will not be complete i.e. theorems that are true that cannot be proven within the system. But this only applies to systems dealing with the arithmetic of natural numbers, there are plenty of systems that can be shown to be consistent and complete. This is a rabbit hole of deep mathematical logic and questions about the foundations of math. There's a reason Gödel is typically considered one of the greatest logicians ever, up there with Aristotle himself. This is all shit that will take years to learn, but fascinating nonetheless.
>self referenceHe used a variation of the diagonalization method pioneered by Cantor to arrive at his incompleteness results. The same method was also used by Turing in his undecidability results. So there are deep connections between Gödel's results and computability. I'm not an expert on it all but I have studied basic results to an extent. Been meaning to read more when I have some free time.
