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