CAP tells us that databases are fundamentally broken, Godel theorem tells us that math is broken. But besides being important theoretical points, they don't matter that much in practice. In the same way, the two generals problem doesn't prevent TCP from working fine.
Gödel's incompleteness (that, by a diagonalization argument, formal systems of minimal expressivity will contain true statements that cannot be proven) applies and will continue to apply to all such formal systems. The bar is so low that any interesting formal system is affected by it. Also, it does not make math broken at all - those formal systems can and in fact must be consistent for the proof to go through (1st incompleteness theorem), and it's just a pity that whatever formal system one comes up with, if it contains arithmetic, it won't be able to prove it's own consistency (2nd incompleteness theorem).
In the words of von Neumann: Gödel's two past papers ("Widerspruchfreiheit" and Continuum Hypothesis), are in any case worth more than the total literary output, past, present and future, of most mathematicians [...]