User blog comment:Wythagoras/Dollars function/@comment-1605058-20130707063642/@comment-1605058-20130707071858

If I is really inaccessible cardinal, you are wrong. It is easy to see that limit of your sequence has cofinality w (it follows from that sequence itself has length w). For cardinal to be inaccessible, it must be regular, i.e. equal to its cofinality. ZFC can prove existence of successor and singular (non-regular) cardinals, but not limit regular ones.