Minor note- When trying to prove Strong Foundationalism (on which I have since given up), I came up with the idea of founding logic not on something anybody must accept but something that must be true in any possible universe. (E.g 1+1=2 according to traditional logic, reductionism if I understand Elizier correctly). This gets around the tortoise's problem and reestablishes logic.

Of course, this isn't so relevant because the tortoise can in response suggest the possibility Achilles is insane either in his reasoning or his memory (or both, but that's superflous) being so far off-track that he can't trust them to perform proper reasoning.