TITLE: Is the set of non finitely describable real numbers closed under addition and squaring?
QUESTION [5 upvotes]: Is the set of non finitely describable real numbers closed under addition and squaring?  If so, can someone give a proof?  Thanks.

REPLY [2 votes]: if $x$ is not finitely describable, then $-x$ is not finitely describable either. But $0=x+(-x)$ is finitely describable so no closure under addition. 
Now if $x^2$ is finitely describable, then $x$ is finitely describable (it's described as "the square root (either positive or negative) of the number described by the description of $x$"). So, you do have closure under squaring.