On Monotone Presentations of Borel Sets

Abstract

If $A$ is a ${\bf \Sigma}^{0}_{\xi}$ set and $A_{n}$ $(n \omega)$ are Borel sets then we call $\{A_{n} \colon n \omega\}$ a presentation of $A$ if $A = \bigcup_{n \omega}A_{n}$ and $A_{n}$ $(n \omega)$ have lower Borel class than $A$ has. We show that for $2 \leq \xi \omega_{1}$ it is not possible to assign a presentation to ${\bf \Sigma}^{0}_{\xi}$ sets in a monotone way; i.e., it is not possible to define functions $f_{n} \colon {\bf \Sigma}^{0}_{\xi} \rightarrow {\bf \Pi}^{0}_{\xi}$ $(n \omega)$ such that for every $A \in {\bf \Sigma}^{0}_{\xi}$ we have $A = \bigcup_{n \omega}f_{n}(A)$ and $A, A' \in {\bf \Sigma}^{0}_{\xi}$, $A \subseteq A'$ implies $f_{n}(A) \subseteq f_{n}(A')$ $(n \omega)$. This answers a question of M\'arton Elekes in the negative. We also show the nonexistence of monotone presentation for Borel functions.