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

TL;DR

This paper investigates the properties of relativized Chaitin's Omega within set theory, showing non-injectivity and degree invariance issues, and explores the complexity of functions mapping to random reals under certain axioms.

Contribution

It proves non-injectivity and degree invariance failure of relativized Omega under ZF, and characterizes functions to random reals under ZF+AD.

Findings

Relativized Omega is not injective on pointed sets under ZF.

Omega fails to be degree invariant in a strong sense.

Functions to x-random reals are uncountable-to-one over an upper Turing cone.

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.