User blog comment:Vel!/1+w/@comment-2033667-20141011224818

I think a good way of getting around the problem would be to very clearly delineate ordinals from a different kind of construction called a "fundamental sequence expression" (FSE). There is be a many-to-one mapping between FSE's and ordinals. For example, "1+w" (with quotes to distinguish it from an arithmetic expression) would be a sort of function that outputs 1, 2, 3, 4, ... With FSE's we can distinguish f_"1+w" from f_"w".

Pedagogically, you can also introduce FSE's after introducing pure ordinals so that the intermixing causes a minimum of confusion.