[FOM] HELP WITH ORDINAL NOTATIONS

[laureano luna]

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?
   
  Regards.
   
  Laureano.
   

		
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