A CAPTCHA is supposed to be a task that is easy for humans, but difficult for computers. High-level mathematics is difficult for humans but easy for computers.
They do have some success of telling computers and humans apart, though.
Stops those pesky humans wasting precious robot bandwidth and is quite a bit harder than the Quantum Random Bit Generator captcha ;)
If you're thinking of either of Gödel's incompleteness theorems, you may be slightly mistaken about what they say. In general, a human operating by rigorous standards of proof is no more able to prove the completeness and consistency of certain formal systems—using the tools provided by those formal systems—than a machine can.
If, as a human, you somehow prove the consistency of these particular formal systems, you run smack into Gödel's second theorem: For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent. Thus, any proof of consistency is self-defeating, whether it's made by neurons or silicon.
Really, math doesn't care what parts of the periodic table you use to prove things. :-)