[FOM] Closure ordinal for Kripke construction?

[Chris Scambler]

In the paper "Outline for a Theory of Truth", Kripke states without proof
that the Church-Kleene ordinal is the closure ordinal for his fixed point
construction over the standard model of arithmetic.

(1) Is there a published proof of this?
(2) Is the result known to transfer to arbitrary structures M (as in, the
closure ordinal for the Kripke construction over M is the least admissible
over M)?

cheers

C
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