[FOM] HELP WITH ORDINAL NOTATIONS
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?
Regards.
Laureano.
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