Show that the set of incompressible strings contains no infinite subset that is Turing$-$recognizable. Proof: Assume to the contrary that an infinite set of incompressible strings is Turing$-$recognizable. Then we can construct TM M which on input n finds a string of length n which is incompressible and outputs it$.$ However, for large n$,$ that gives a short description of the incompressible string, a contradiction.To compute the descriptive complexity of strings K$($x$)$ from x with an oracle for ATM, run all descriptions of length $|$x$|+$ c where c is the constant in Theorem $6\mathrm{.}27,$ using the ATM oracle to avoid those which halt. The length of the shortest is K$($x$).$