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

Abstract

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-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.

arXiv commentWe added a new result and fixed some typos