  How should one deal with the complications introduced by the ramified hierarchy? Russell showed, in the introduction to the second edition to  Principia Mathematica,  that these complications can be avoided in some cases. He even thought, in Appendix B of the second edition of Principia Mathematica, that the ramified hierarchy of natural numbers of order 1,2,… collapses at order 5 for defining the reflexive transitive closure of a relation. However, Gödel later found a problem in his argument, and indeed, it was shown by Myhill 1974 that this hierarchy actually does not collapse at any finite level. A similar problem, discussed by Russell in the introduction to the second edition to  Principia Mathematica  arises in the proof of Cantor’s theorem that there cannot be any injective functions from the collection of all predicates to the collection of all objects (the version of Russell’s paradox in Frege’s system that we presented in the introduction). Can this be done in a ramified hierarchy? Russell doubted that this could be done within a ramified hierarchy of predicates and this was indeed confirmed indeed later (see Chwistek 1926, Fitch 1939 and Heck 1996). 