[justinpombrio]

> by assuming that a system that can be formally modeled can prove something that cannot be proven within a formal system

Something that cannot be proven within which formal system? Every true statement can be proven by some formal system. No formal system can prove all true statements.

(This is all about Godel incompleteness. Turing incomputability does apply though, see the sibling comments! It's important to keep them separate, they're saying different things.)