For formal logic, the consequences are dire. For mathematics, they are negligible.
A system of formal logic is a consistent (contradiction-free) recursively defined set whose members are rules of inference. Godel’s theorem shows that there is no consistent, recursively defined class contains every arithmetical truth and, consequently, that any give system of formal logic is so weak that that there exist inferences licensed by elementary arithmetic that fall outside its scope (inferences such as ‘if Jim has three apples and Mary has four apples, and they have no apples in common, then Jim and Mary jointly have seven apples’). This shows that the power of any given system of logic is pathetically limited.
Mathematics is the study of hypothetical structures, and it is therefore the class of truths of the form: ‘if x is a structure of such and such a type, then thus and such is also true of x.’ A consequence of Godel’s theorem is that there is no way of enumerating the class of mathematical truths. But this has no bearing on their legitimacy, since mathematical statements are either true or false in virtue of the structures they describe, and not in virtue of their being outputs of some recursion.