Given an undecidable collection of first-order sentences, is there necessarily a complete undecidable theory containing it? A direct attempt to prove it seems to require some control over the completions which need not exist, but I don't see a counterexample.


On a more concrete side, Macyntire proved that the theory of all pairs of real closed fields is undecidable. I am interested to know if there is a particular pair of real closed fields with undecidable theory.



Undecidable completion of undecidable theory, and pairs of RCF
My recollection is that Macintyre proved there are $2^{\aleph\_0}$ complete theories of pairs
$(K,L)$ where $L\subset K$ are real closed fields. This is in his thesis but I don't think
he published it anywhere else. There are later papers of Francoise Delon and Walter Bauer
that develop this further. On the other hand there is an earlier theorem of Robinson's that
if $L\subset K$ are real closed and $L$ is dense in $K$ then the theory of $(K,L)$ is decidable. 


