User blog comment:Wythagoras/All my stuff/@comment-10429372-20130715145943/@comment-5529393-20130715190215

Dr. Ceasium, the iota function cannot take every other defined function in its function set, since for example we can define a function like iota(n)+1, and we would get iota(n) > iota(n)+1, a contradiction. So we need to restrict which functions can go in the function set. Assuming that we don't include functions that refer to all definable functions in the function set, then iota_m(n) will be a definable function for all m, as it will be derived from finitely many definable functions and hence be definable itself. So the ¥ function will exceed iota_m (n) for any m.

LittlePeng9:  JDH doesn't assume that V=HOD or V=L. In fact he proves that any countable model of ZFC can be extended to a pointwise definable model.