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By a theorem of Sacks, if a real $x$ is recursive relative to all elements ofa set of positive Lebesgue measure, $x$ is recursive. This statement, and theanalogous statement for non-meagerness instead of positive Lebesgue measure,have been shown to carry over to many models of transfinite computations. Here,we start exploring another analogue concerning recognizability rather thancomputability. We introduce a notion of relativized recognizability and showthat, for Infinite Time Turing Machines (ITTMs), if a real $x$ is recognizablerelative to all elements of a non-meager Borel set $Y$, then $x$ isrecognizable. We also show that a relativized version of this statement holdsfor Infinite Time Register Machines (ITRMs). This extends our earlier workwhere we obtained the (unrelativized) result for ITRMs. We then introduce ajump operator for recognizability, examine its set-theoretical content and showthat the recognizable jumps for ITRMs and ITTMs are primitive-recursivelyequivalent, even though these two models are otherwise of vastly differentstrength. Finally, we introduce degrees of recognizability by considering thetransitive closure of relativized recognizability and connect it with therecognizable jump operator to obtain a solution to Post's problem for degreesof recognizability.

Infinite Time Recognizability from Random Oracles and the Recognizable Jump Operator