# [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

My error is probably obvious but I can’t find it out. Could anyone help me?

Regards.

Laureano.

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