| Sumario: | We study genericity and randomness with respect to ITTMs, continuing the work initiated by Carl and Schlicht. To do so, we develop a framework to study randomness in the constructible hierarchy. We then answer several of Carl and Schlicht's question. We also ask a new question one the equality of two classes of randoms. Although the natural intuition would dictate that the two classes are distinct, we show that things are not as simple as they seem. In particular we show that the categorical analogues of these two classes coincide, in contradiction with the natural intuition. Even though we are not able to answer the question for randomness in this article, we delineate and sharpen its contour and outline.
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