I would like to see a proof or a sketch of a proof that the set of incompressible strings is undecidable.
Definition: Let x be a string, we say that x is c-compressible if K(x) $\leq$ |x|-c. If x is not c-compressible, we say that x is incompressible by c. K(x) represents the Kolmogorov complexity of a binary string x.
Theorem: incompressible strings of every length exist.
Proof: The number of binary strings of length n is $2^{n}$, but there exist $\displaystyle\sum\limits_{i=0}^{n-1} 2^i$=$2^{n}$-1 descriptions of lengths that is less than n. Since each description describes at most one string then there is at least one string of length n that is incompressible.
From here I feel it is natural to ask whether or not the set of incompressible strings is decidable, the answer is $\textit{no}$, but I would like to see the justification via a proof or proof sketch.
Edit: I would like to add I am already familiar/comfortable with the proof that Kolmogorov complexity is uncomputable.