See: https://www.sciencedirect.com/science/article/pii/S0019995884800601 https://www.sciencedirect.com/science/article/pii/S001999588... look at section 1.2 i

See: https://www.sciencedirect.com/science/article/pii/S0019995884800601 https://www.sciencedirect.com/science/article/pii/S001999588... look at section 1.2 it is the non growth theorem for algorithmic information, but also deals with randomness Levin proves randomness + computation is not expected to increase algorithmic mutual information. In other words there is no randomness loophole in Godël's theorem.