Some more results on relativized Chaitin's $\Omega$

Liang Yu

We prove that, assuming $\mathrm{ZF}$, and restricted to any pointed set, Chaitin's ${\Omega }_U:x\mapsto {\Omega }_U^x={\sum }_{U^x(\sigma )\downarrow }{2}^{-|\sigma |}$ is not injective for any universal prefix-f…