laureano luna laureanoluna at yahoo.es
Fri Mar 10 16:26:53 EST 2006

I need help with the following.
  Torkel Franzen introduces (in Inexhaustibility pp.156-162) a set O of arithmetized notations for ordinals below omega-1-CK.
  On page 189 he writes that by Kleene's second recursion theorem (p. 155) there is a function index e such that:
  1. {e}(0) = suc(lim(e))
  where “{e}(0)” and “suc(lim(e))” are numbers/notations in O (p.157) and {e}denotes the function whose index is e. 
  But from 1. we get:
  2. ï{e}(0)ï = ïsuc(lim(e))ï
  where, if a is an ordinal notation, then ïaï is the ordinal denoted by a.                                    
  We also have:
  3. {e}(0) o< lim(e) o< suc(lim(e))
  where o< is a transitive partial ordering recursively defined for ordinal notations in O (p. 157). 
  And thus:
  4. {e}(0) o< suc(lim(e))
  Hence (p. 158):
  5. ï{e}(0)ï < ïsuc(lim(e))ï
  in contradiction with 2.
  My error is probably obvious but I can’t find it out. Could anyone help me?


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