The Tortoise's mind needs the dynamic of adding Y to the belief pool when X and (X→Y) are previously in the belief pool. If this dynamic is not present—a rock, for example, lacks it—then you can go on adding in X and (X→Y) and (X⋀(X→Y))→Y until the end of eternity, without ever getting to Y.

This appears to be a circular argument.

Maybe we could honestly accept than impossible demands of rigor are indeed impossible. And focus on what is possible.

This is why I wrote this:

I understand that this has no decision-making value.