I'm interested in examples of theorems that employ the proof techniques that are utilized in the proof of the undecidability of Kolmogorov Complexity.
Definition:(Sipser) Let x be a binary string. We say that the minimal description of x, written as d(x), is the shortest string $\langle$M,w$\rangle$ where TM M on input w halts with x on its tape. So, the Kolmogorov Complexity K(x) is written as, K(x)=|d(x)|. K(x) is defined to be the length of minimal description of x.
Theorem: K(x) is not a computable function.
Proof/Sketch of Proof (attributed to Chor):
Proof of negation. $\forall$n, let $y_{n}$ be the lexicographical first string y that satisfies n < K(y).
Consider the following TM M:
On input n (encoded in binary), M generates one by one all binary strings $x_{0}$, $x_{1}$, $x_{2}$, $x_{3}$… in lexicographic order.
For each $x_{i}$ it produces, M computes K($x_{i}$).
If K($x_{i}$) > n, then the TM M, outputs $x_{i}$ and halts.
Else, the TM M, continues to examine the next lexicographical string $x_{i+1}$.
Since the function K is unbounded, it is guaranteed that M will eventually come across a string x satisfying K(x) $>$ n.
Question: what will the TM M output on input n?
By definition on input n TM M outputs $y_{n}$ (the lexicographical first string whose Kolmogorov complexity exceeds n, K(x) > n), but the length of n is $log_{2}$(n).
So we have $K_{M}$($y_{n}$) $\leq$ $log_{2}$(n). There is a constant $c_{M}$ such that $\forall$y, K(y) $\leq$ $K_{M}$(y) + $c_{M}$, so $\forall$n K($y_{n}$) $\leq$ $log_{2}$(n) + $c_{M}$.
By definition of $y_{n}$ for all n, n < K($y_{n}$). By combining the two inequalities we get: n < $log_{2}$(n) + $c_{M}$, but for large enough n this is false. Thus a contradiction.
Question: What other theorems utilize a similar proof technique in their proofs?
For example:
The proof that the set of incompressible strings is undecidable is very similar with some slight modifications.
Best Answer
Using the same technique, one can construct infinitely many statements which are true with probability arbitrarily close to 1, but are nonetheless unprovable. See lemma 4 in http://theory.stanford.edu/~trevisan/cs172/notek.pdf