[FOM] counted sets

Aatu Koskensilta Aatu.Koskensilta at uta.fi
Tue Aug 25 12:05:07 EDT 2009

Quoting Bill Taylor <W.Taylor at math.canterbury.ac.nz>:

> As I noted, the set of all recursive ordinals is necessarily non-recursive!

Well, it's not just non-recursive, it's Pi-1-1 complete.

> There can be no effective notation for it, though there can be for any lesser
> ordinal.  The thing is, there is no "uniform" way of notating the ordinals
> less than it - every so often one must adopt a whole new procedure,
> largely unrelated to but engulfing the prior ones.

Sure there is, Kleene's O. Of course, what you mean is that there's no  
/univalent/ ordinal notation system for all ordinals < omega_1^CK,  
while there is for any ordinal < omega_1^CK. (A notation system is  
univalent if notations are unique.)

> Hartley Davidson's book has a nice little section on it.

Surely you mean Hartley Rogers?

Aatu Koskensilta (aatu.koskensilta at uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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