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The origin of the uncertainty inherent in quantum measurements has been discussed since quantum theory's inception, but to date the source of the indeterminacy of measurements performed at an angle with respect to a quantum state's preparation is unknown. Here I propose that quantum uncertainty is a manifestation of the indeterminism inherent in mathematical logic. By explicitly constructing pairs of classical Turing machines that write into each others' program space, I show that the joint state of such a pair is determined, while the state of the individual machine is not, precisely as in quantum measurement. In particular, the eigenstates of the individual machines appear to be superpositions of classical states, albeit with vanishing eigenvalue. Because these "classically entangled" Turing machines essentially implement undecidable "halting problems", this construction suggests that the inevitable randomness that results when interrogating such machines about their state is precisely the randomness inherent in the bits of Chaitin's halting probability. Because this classical construction mirrors quantum measurement, I argue that quantum uncertainty has the same origin.