Description from http://www.scottaaronson.com/democritus/lec10.5.html:
Years ago I parodied Penrose’s argument by means of the Gödel CAPTCHA. Recall from Lecture 4 that a CAPTCHA (Completely Automated Public Turing Test to tell Computers and Humans and Apart) is a test that today’s computers can generate and grade, but not pass. These are those “retype the curvy-looking nonsense word” deals that Yahoo and Google use all the time to root out spambots. Alas, today’s CAPTCHA’s are far from perfect; some of them have even been broken by clever researchers.
By exploiting Penrose’s insights, I was able to create a completely unbreakable CAPTCHA. How does it work? It simply asks whether you believe the Gödel sentence G(F) for some reasonable formal system F! Assuming you answer yes, it then (and this is a minor security hole I should really patch sometime) asks whether you’re a human or a machine. If you say you’re a human, you pass. If, on the other hand, you say you’re a machine, the program informs you that, while your answer happened to be correct in this instance, you clearly couldn’t have arrived at it via a knowably sound procedure, since you don’t possess the requisite microtubules. Therefore your request for an email account must unfortunately be denied.