Coding is non-robust

Abstract

For various reasons, higher-order objects are often studied in mathematical logic via second-order `codes' or `representations'. An important example hailing from real analysis and topology is provided by open sets, which are represented by unions $\cup_{n\in \mathbb{N}}I_{n}$ where each $I_{n}$ is a basic open interval. Now, Montalbán has recently highlighted the importance of robustness of logical systems in Reverse Mathematics (abbreviated RM). It is then a natural RM-question whether basic properties of open sets are robust under slight modifications of the coding of open sets. Here, we study the representation where $I_{n}$ as above is \emph{either} an open interval \emph{or} the union of two such intervals \emph{but} we cannot decide which one. Under this slight variation of the usual coding, basic properties of open and closed sets readily imply the relatively strong system ATR$_{0}$ from RM. Moreover, we obtain equivalences for the former properties and the \emph{enumeration principle}. The latter states that countable sets can be enumerated and boasts many equivalences from Fourier analysis. Along the way, we investigate the RM-properties of the closure, interior, and boundary of sets of reals, a study interesting in its own right.