@dmg Just because the possible sequences have a well defined order does not mean you can iterate through them to select one. The real numbers have a well defined order, but they are not countable.

@dmg So you are saying that the number of classes is uncountable but the members of a single class is countable right? I think Steve's problem is with selecting the class. Throwing the axiom of choice at it is fine in theory, but in practice how would you identify the class?

While the members of a particular class are countable and well-ordered, they don't have any natural reference point: they're defined relative to an uncountable set. Any member of a class has an infinite number of values both before and after it. The first element in lexicographical order is going to be the equivalent of negative infinity (but we can't determine where zero is).