We know that Kolmogorov Cmplexity (KC) has connections to mathematical logic since it can be used to prove the Gödel incompleteness results (Chaitin's Theorem and [Kritchman-Raz][1]). Are there any other striking application of Kolmogorov complexity to mathematical logic (outside KC itself of course)?

Relatively simple examples like the ones I mentioned are preferred, but more complicated illustrations are also very welcome!


(This is cross posted from [MSE][2], because I wasn't getting any replies there)


  [1]: https://arxiv.org/abs/1011.4974
  [2]: https://math.stackexchange.com/questions/3742524/the-connection-between-kolmogorov-complexity-and-mathematical-logic