论文标题
相对化的Chaitin的$ω$的更多结果
Some more results on relativized Chaitin's $Ω$
论文作者
论文摘要
我们证明，假设$ \ mathrm {zf} $，并且仅限于任何指向的集合，Chaitin的$ω_U：x \ mapstoω_u^x = \ sum_ {u^x（σ）\ d odarow;从非常强烈的意义上讲，学位不变，回答了描述性集理论中最近的几个问题。此外，我们表明，在$ \ mathrm {zf}+\ mathrm {ad} $下，每个函数$ f $ pu $映射$ x $ to $ x $ - random必须在图灵学位的上锥上不可容纳。
We prove that, assuming $\mathrm{ZF}$, and restricted to any pointed set, Chaitin's $Ω_U:x\mapsto Ω_U^x=\sum_{U^x(σ)\downarrow}2^{-|σ|}$ is not injective for any universal prefix-free Turing machine $U$, and that $Ω_U^x$ 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.
