Sparse-witness construction of a simple set
= Sparse-witness construction of a simple set
Enumerate all <computably enumerable sets> as $W_e$. For each $e$, once a witness $x>2e$ appears in $W_e$, enumerate one such witness into $X$ and never act for $e$ again. Every infinite $W_e$ supplies a witness. Only requirements $e<n$ can put numbers into $[0,2n]$, so at least $n+1$ numbers in that interval stay outside $X$. Thus $X$ is a <simple set>. The proof uses a numerical bound rather than an infinitude test.