# A countable ordinal definable set of reals without ordinal definable elements

## Victoria Gitman

### The City University of New York

In 2010, a question on MathOverflow asked whether it is possible that a countable ordinal definable set of reals has elements that are not ordinal definable. It is easy to see that every element of a finite ordinal definable set of reals is itself ordinal definable. Also it is consistent that there is an uncountable ordinal definable set of reals without ordinal definable elements. It turned out that the question for countable sets of reals was not known. It was finally solved by Kanovei and Lyubetsky in 2014, who showed, using a forcing extension by a finite-support product of Jensen reals, that it is consistent to have a countable ordinal definable set of reals without ordinal definable elements. In the talk, I will give full details of their argument.

An extended abstract is available on my blog here.

Victoria Gitman received her Ph.D. in 2007 from the CUNY Graduate Center, as a student of Joel Hamkins, and is presently a visiting scholar at the CUNY Graduate Center. Her research is in Mathematical Logic, in particular in the areas of Set Theory and Models of Peano Arithmetic.