The Tortoise assumes a proposition \(p\) and a material conditional \(p \implies q\).
The exact \(p\) and \(q\) aren’t important to the moral of the story, though it’s something like “If \(A=B\) and \(B=C\) (\(p\)), then \(A=C\) (\(q\))”
The Tortoise is playing a game: I’ll do anything you tell me to do, so long as you make explicit the rule you’re asking me to follow.
Achilles tries to convince the Tortoise to accept \(q\).
He says that logic obliges you to acknowledge \(q\) in this case.
The Tortoise is willing to go along with this but demands that this rule be made explicit:
Achilles adds an extra axiom: \(p \land (p \implies q) \implies q\).
Achilles says that, now, you really have to accept \(q\), given that you’re committed to:
\(p\)
\(p \implies q\)
\(p \land (p \implies q) \implies q\).
But the Tortoise notes that, if taking those three propositions and concluding \(q\) is really something logic obliges one to do, then it bears writing down:
\(p \land (p \implies q) \land (p \land (p \implies q)) \implies q\)
This can go ad infinitum; the Tortoise wins.