[FOM] Ordinal definable numbers

[Ali Enayat]

About two months ago (June 10), Tim Chow posed the following question
to FOM subscribers, a question that was originally posed by Garabed
Gulbenkian on MathOverflow.net, see

http://mathoverflow.net/questions/17608/a-question-about-ordinal-definable-real-numbers).

>Is the following statement consistent with ZFC?

>"There exists a denumerably infinite and ordinal definable set of real
>numbers, not all of whose elements are ordinal definable."

The point of this posting is twofold:

(1) The statement "There exists a definable set of real numbers that
is Dedekind finite, but none of whose members is ordinal definable" is
consistent with ZF (not ZFC!); and

(2) To suggest a model of ZFC that will very likely yield a positive
answer to Garabedian's original question.

(1) follows from combining two results - one by Solovay, the other by
Kanovei . Both concern the symmetric submodel N of the generic
extension M[G] corresponding to adding countably many side-by-side
"Jensen generic reals" to a model M of ZFC.

For an expository account of Jensen-generic reals see Section 28 of
Jech's Set Theory (Millenium edition).  Here let me just say that
Jensen reals are Delta^1_3 definable and minimal. The notion of
forcing that produces them is obtained by "thinning" Sacks forcing
using the diamond principle.

Solovay showed that the collection of Jensen reals in N is definable.
Kanovei, on the other hand, observed that the same collection is
Dedekind finite.

Solovay's proof appears in the following paper of mine as Theorem 3.3.

On the Leibniz-Mycielski Axiom in Set Theory , Fundamenta
Mathematicae, vol. 181 (2004), pp.215-231.

Now regarding (2). The model is very likely to yield a positive answer
to the original question is the generic extension M[G] corresponding
to adding countably many side-by-side "Jensen generic reals" to a
model M of ZFC.

Regards,

Ali Enayat