Do stronger definitions of randomness exist?

In this paper, we investigate refined definition of random sequences. Classical definitions (Martin-Löf tests of randomness, uncompressibility, unpredictability, or stochasticness) make use of the notion of algorithm. We present alternative definitions based on set theory and explain why they depend on the model of ZFC that is considered. We also present a possible generalization of the definition when small infinite regularities are allowed.