Is the set of non finitely describable real numbers closed under addition and squaring? Is the set of non finitely describable real numbers closed under addition and squaring?  If so, can someone give a proof?  Thanks.

Hum... if by non-finitely-describable you mean "can't construct a finite description (as e.g. a Turing machine)", they aren't closed with respect to addition: $a + b = 2$ if $a$ is one of yours, $b$ is too (if it wasn't, $a$ could be described). But 2 clearly isn't.
Squares I have no clue.
