Does Gödel’s incompleteness theorem have implications beyond mathematics? Is it a worm in the apple of rationality?

No. Absolutely no one should have ever been surprised that mathematical truth cannot be equated with theoremhood in some finite axiomatic system. An infinitude of mathematical truths are uninteresting trivia, with no obvious route to being proved. Example: let’s say that the decimal expressions of the square root of 17 and pi to the 27th power “match” just in case either they have the same digit in the tenths place, or the same two digits in the next two places, or the same three digits in the next three places, etc. If we treat these decimal expressions as essentially random sequences of digits, then the a priori chance that these two numbers match is one out of nine.

Now: how do we tell if they match or not? Well, we can just calculate out the sequences of digits and check. And if they match we will eventually find the match and prove that they match. But what if, as is likely, they don’t match? No amount of just grinding out the digits and checking will ever prove it: there are always more digits to check. And I see zero prospect of any other way to prove that they don’t match. So if they don’t match, that is an unprovable mathematical fact. It is also a very, very, very uninteresting one. All Gödel did was find a clever way to construct a provably unprovable mathematical fact, given any consistent and finite set of axioms to work with. The work is clever but in no way profound. It should have come as no surprise at all.