# [FOM] 508: Unique Undefinable Elements

Harvey Friedman hmflogic at gmail.com
Sat Dec 22 20:07:48 EST 2012

Thanks for the reference. I didn't know about this problem of Arnie
Miller that Hjorth solved there! You can find the paper on the
Internet through Google, searching for     Unique Undefinable Elements
Hjorth     and opening up the second hit.

Miller (poser) and Hjorth (solver), using the infinitary language
L_omega1,omega get a result that is grammatically almost identical, to
mine, but of a very different character - e.g., in the following
senses.

1. No such Hjorth structure can be countable.
2. Hjorth's structure has cardinality omega_1.

For the ordinary predicate calculus problem I did,

1. All such structures must be countably infinite.
2. My structure is recursive in the omega jump.

Also, there does not appear to be any relationship between the
technical arguments.

For both kinds of problems, I think there is a lot of further research
possibilities.

Harvey Friedman

For my original postings, see
http://www.cs.nyu.edu/pipermail/fom/2012-November/016821.html

On Tue, Dec 11, 2012 at 2:04 AM, Andrew Brooke-Taylor
<andrewbt at gmail.com> wrote:
>
> Dear FOMers,
>
> I just wanted to mention to readers that similarly, Greg Hjorth
> constructed a model with a unique element which is not definable in
> L_{\omega_1, \omega} (JSL 75 no. 1 (2010), pp 269-274.  Sorry, I'd
> make it a link, but I only have hard copy access here).
>
> Best wishes,
> Andrew Brooke-Taylor
>
>