[FOM] Closure ordinal for Kripke construction?

[Yiannis N. Moschovakis]

​​
(2) is correct, that on any structure M (that has an elementary
ordered pair), the closure ordinal of the Kripke construction is
the closure ordinal of M, which is the same as the ordinal of the
smallest admissible set (with atoms) above M.

I don't know of an easily accessible, published, short and direct
proof of this, but it follows easily from the remarks in Kripke's
paper (especially Footnote 24 and the last paragraph) and standard
facts about inductive definability. The relevant references are
my 1974 book on "Elementary induction on abstract structures",
Barwise's "Admissible sets and structures", and (especially) my
paper on "Sense and denotation as algorithm and value" which
discusses explicitly the connection between Kripke's language and
inductive definability. (This is posted on my homepage.)

Yiannis Moschovakis


On Tue, Nov 21, 2017 at 6:05 AM, 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?
> (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
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> https://cs.nyu.edu/mailman/listinfo/fom
>
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20171121/5bb45955/attachment.html>