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Aczel bn+1 bracket symbol cancellation laws Cantor normal form cardinal chapter classical ordinals Clearly completes the proof contains an infinite contradiction converse corollary Crossley definition denote directed refinement theorem E-numbers embedded enumeration equivalent follows at once hence by theorem Immediate from theorem implies infinite r.e. subset initial segment integers isolated Kleene last element least upper bound leave the reader limit number linearly ordered sets losols minimal upper bounds natural numbers Nerode notation number for addition number for exponentiation numbers for multiplication obtain one-one partial recursive order embedding order isomorphism order preserving ordinal sum Parikh partial recursive function principal numbers proof of theorem prove quasi-finite C.O.T.s recursive descending chain recursive function defined recursive isomorphism recursive ordinals recursive quord recursive set refinement theorem 2.3.2 sequence number Sierpinski similarly successor number Suppose Theorem cf transfinite induction undefined otherwise well-ordering of type write