Some more results on relativized Chaitin's Ω

Abstract

We prove that, assuming ZF, and restricted to any ${\le }_T$-pointed set, Chaitin's ${\Omega }_U:x\mapsto {\Omega }_U^x={\sum }_{U^x\left(\sigma \right)\downarrow }{2}^{-\left|\sigma \right|}$ is not injective for any universal prefix-free Turing machine U, and that ${\Omega }_U^x$ fails to be degree invariant in a very strong sense, answering several recent questions in descriptive set theory. Moreover, we show that under $\mathrm{ZF}+\mathrm{AD}$, every function f mapping x to x-random must be uncountable-to-one over an upper cone of Turing degrees.