[FOM] Closure ordinal for Kripke construction?

[Benedict Eastaugh]

> On 21 Nov 2017, at 06:05, Chris Scambler <cscambler at gmail.com> wrote:
> 
> 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?

Yes, for example in Meadows (2015), corollary 5.23.

Meadows, T. Infinitary tableau for semantic truth. The Review of Symbolic Logic 8(2):207–235, June 2015. https://doi.org/10.1017/S175502031500012X

> (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)?

I don’t know the answer to this, but I would guess that the answer is yes if M is an omega-model.

Best,
Benedict