If ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) is consistent, does it remain consistent
when the following statement is added to it as a new axiom? 


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


If the answer to the above question is negative, then it must be provable in ZFC that every denumerably
infinite and ordinal definable set of real numbers is hereditarily ordinal definable. This is because
every real number can be regarded as a set of finite ordinal numbers and every finite ordinal number is
ordinal definable.
 Garabed Gulbenkian



A question about ordinal definable real numbers
The original problem solves in the positive: there is a model of ZFC in which there exists a countable OD (well, even lightface $\Pi^1\_2$, which is the best possible) set of reals $X$ containing no OD elements. The model (by the way, as conjectured by Ali Enayat at <http://cs.nyu.edu/pipermail/fom/2010-July/014944.html>) is a $\mathbf P^{<\omega}$-generic extension of $L$, where $\mathbf P$ is Jensen's minimal $\Pi^1\_2$ real singleton forcing and $\mathbf P^{<\omega}$ is the finite-support product of $\omega$ copies of $\mathbf P$. 


**A few details.** Jensen's forcing is defined in $L$ so that $\mathbf P =\bigcup\_{\xi<\omega\_1} \mathbf P\_\xi$, where each $\mathbf P\_\xi$ is a ctble set of perfect trees in $2^{<\omega}$, *generic* over the outcome $\mathbf P\_{<\xi}=\bigcup\_{\eta<\xi}\mathbf P\_\eta$ of all earlier steps in such a way that any $\mathbf P\_{<\xi}$-name $c$ for a real ($c$ belongs to a minimal countable transitive model of a fragment of ZFC, containing $\mathbf P\_{<\xi}$), which $\mathbf P\_{<\xi}$ forces to be different from the generic real itself, is pushed by $\mathbf P\_{\xi}$ (the next layer) not to belong to any $[T]$ where $T$ is a tree in $\mathbf P\_{\xi}$. The effect is that the generic real itself is the only $\mathbf P$-generic real in the extension, while examination of the complexity shows that it is a $\Pi^1\_2$ singleton. 


Now let $\mathbf P^{<\omega}$ be the finite-support product of $\omega$ copies of $\mathbf P$. It adds a ctble sequence of $\mathbf P$-generic reals $x\_n$. A version of the argument above shows that still the reals $x\_n$ are the only $\mathbf P$-generic reals in the extension and the set $\{x\_n:n<\omega\}$ is $\Pi^1\_2$. Finally the routine technique of finite-support-product extensions ensures that $x\_n$ are not OD in the extension.


**Addendum.** For detailed proofs of the above claims, see [this manuscript](http://arxiv.org/abs/1408.3901).


Jindra Zapletal informed me that he got a model where a $\mathsf E\_0$-equivalence class $X=[x]\_{E\_0}$ of a certain Silver generic real is OD and contains no OD elements, but in that model $X$ does not seem to be analytically definable, let alone $\Pi^1\_2$. The model involves a combination of several forcing notions and some modern ideas in descriptive set theory recently presented in *Canonical Ramsey Theory on Polish Spaces*. Thus whether a $\mathsf E\_0$-class of a non-OD real can be $\Pi^1\_2$ is probably still open.


**Further Kanovei's addendum of Aug 23**. 
It looks like a clone of Jensen's forcing on the base of Silver's (or $\mathsf E\_0$-large Sacks) forcing instead of the simple Sacks one leads to a lightface $\Pi^1\_2$ generic $\mathsf E\_0$-class with no OD elements. The advantage of Silver's forcing here is that it seems to produce a Jensen-type forcing closed under the 0-1 flip at any digit, so that the corresponding extension contains a $\mathsf E\_0$-class of generic reals instead of a generic singleton. I am working on details, hopefully it pans out.


**Further Kanovei's addendum of Aug 25**. 
Yes it works, so there is a generic extension $L[x]$ of $L$ by a real in which the 
$\mathsf E\_0$-class $[x]\_{\mathsf E\_0}$ is a lightface $\Pi^1\_2$ (countable) set with no OD elements. I'll send it to Axriv in a few days.


**Further Kanovei's addendum of Aug 29**. [arXiv:1408.6642](http://arxiv.org/abs/1408.6642)


