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 ∪n∈ ℕIn where each In 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 In 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 ATR0 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.
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BibTeX
@misc{sanders2026coding,
title = {{Coding is non-robust}},
author = {Sam Sanders},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18371},
eprint = {2609.18371},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18371},
}
RIS
TY - UNPB
TI - Coding is non-robust
AU - Sam Sanders
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18371
UR - https://arxiv.org/abs/2609.18371
AB - 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 ∪_n∈ ℕI_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.
ER -