User blog comment:Wythagoras/All my stuff/@comment-7484840-20130715121715/@comment-5150073-20130715125251

I guess he meant that outputs of ¥ function can include the number defined in non-strict mahtematical sense.

It's questionable whether both functions will reach the order type $$\omega_1$$. Even though we can make a successor for any ordinal, possibly there exists countable ordinal which is so large that his fundamental sequence diagonalizes over real world.