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

David Asperó¹

¹ Department of Mathematics University of Bristol University Walk Bristol, BS8 1TW, England, UK

Fundamenta Mathematicae, Tome 186 (2005) no. 3, pp. 215-231.

Résumé

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 $\omega _1$).

Détail

DOI : 10.4064/fm186-3-2

Keywords: several results presented concerning existence nonexistence subset omega real which works robust code respect given sequence langle alpha alpha omega rangle pairwise disjoint stationary subsets omega where robustness code may either mean langle ast alpha alpha omega rangle whenever each ast alpha equal alpha modulo nonstationary changes may have weaker meaning langle alpha cap alpha omega rangle every club subseteq omega variants above theme considered which result requirement gets exactly coded replaced weaker requirement set coded which equal club sequences stationary sets replaced decoding devices possibly carrying information functions omega omega