Yeah, it's hard to phrase this well and I don't know if there's a standard phrasing. What I was trying to get at was the idea that some computable ordering is total and well-ordered, and therefore an ordinal.

Well, supposing that a large ordinal exists is equivalent to supposing a form of Platonism about mathematics (that a colossal infinity of other objects exist). So that is quite a large statement of faith!

All maths really needs is for a large enough ordinal to be logically possible, in that it is not self-contradictory to suppose that a large ordinal exists. That's a much weaker statement of faith. Or it can be backed by an inductive argument in the way Eliezer suggests.