## The nonexistence of robust codes for subsets of $\omega _1$

### Volume 186 / 2005

#### Abstract

Several results are presented concerning the existence or nonexistence, for a subset $S$ of $\omega _1$, of a real $r$ which works as a robust code for $S$ with respect to a given sequence $\langle S_\alpha :\alpha <\omega _1\rangle $ of pairwise disjoint stationary subsets of $\omega _1$, where “robustness” of $r$ as a code may either mean that $S\in L[r,\langle S^\ast _\alpha :\alpha <\omega _1\rangle ]$ whenever each $S^\ast _\alpha $ is equal to $S_\alpha $ modulo nonstationary changes, or may have the weaker meaning that $S\in L[r, \langle S_\alpha \cap C : \alpha <\omega _1\rangle ]$ for every club $C\subseteq \omega _1$. Variants of the above theme are also considered which result when the requirement that $S$ gets exactly coded is replaced by the weaker requirement that some set is coded which is equal to $S$ up to a club, and when sequences of stationary sets are replaced by decoding devices possibly carrying more information (like functions from $\omega _1$ into $\omega _1$).