Because of the halting issue you cannot summarise reality into a set of axioms and rules. Which is the original purpose of mathematics. Therefore you cannot say mathematics is right or true without adding at the end: "this actually could all be wrong".
> Because of the halting issue you cannot summarise reality into a set of axioms and rules
Conway's Game of Life is a kind of reality summarized into axioms and rules, and most likely our universe/multiverse is on track to being similarly summarized. The halting issue doesn't get in the way of that achievement...
Once you have the intuition for why the Halting Problem (i.e. fortune teller paradox) is obvious, then Godel's proof is just an XSS attack on axiom systems whose designers think they're only "about numbers", like how CSS is designed to be about styling but it's also Turing-complete and vulnerable to XSS.
Russel and Whitehead were trying to simultaneously (1) invent a system capable of proving highly insightful claims about the infinite space of numbers, like "there doesn't exist a number between 1 and infinity with property P" but (2) not a system that can encode a Turing-equivalent agent that creates paradoxes if it tries to predict its own future.
Godel was just the first to point out that condition (2) was already met just by supporting Peano Arithmetic, the same way a modern computer science undergrad can point out that CSS is Turing-complete or your regex-based HTML validator is vulnerable to XSS attacks.
CSS is not turing-complete - it is as turing-complete as a simple TXT file.
Deciding whether some sequence of bytes resembles a CSS file is very possible.
Deciding whether a stylesheet can be successfully applied to a website is also decidable.
Of course, it is undecidable whether a certain pattern emerges if a stylesheet is applied to an infinitely large HTML file.
But then, it is also undecidable whether some TXT file appears in an infinitely large HTML file.
It's Turing complete if you let a human feed a bunch of clicks to proceed along the computation steps. [1] It also allows embedded JS (in some browsers), which is Turing complete. My point is that languages really quickly surpass the Turing complete barrier if you let them do fancy things. Godel realized that proving interesting statements about unbounded natural numbers is quite a fancy thing.
In this case the XSS vector is the fact that CSS allows running JavaScript in some browsers [1], but still, a lot of systems that are supposed to be weak and domain-specific are discovered to be Turing-complete, e.g. Conway's Game of Life.
Of course proofs can be still correct, it's just that we cannot cope with infinites very well without talking about computational limits.
My argument was hyperbolic, I regret, but it was less about what's incorrect and more about what's incomplete. The aspirations of early mathematicians was to discover a platonic ideal. Instead of that we only got useful tools that are always* up for re-interpretation depending on context.
An example of this is that Euclid's fifth postulate, which be consistent with many other interpretations of geometry too, not just the single one originally intended. It turned into a tool of formal scaffolding instead of an omnipotent "truth". Going from absolute to relative in power.
*In cases where they involve infinites. Including simple expressions like "take N to be an integer"
