[FOM] HELP WITH ORDINAL NOTATIONS

[William Tait]

> 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.
>

The second recursion theorem does not yield that {e} is even a total  
function, much less that lim(e) is in O.

Best regards,

Bill Tait