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It is well known that every Borel set has the property of Baire. That is, for every Borel set $B$, there is an open set $U$ and a sequence of dense open sets $D_n$ such that for every $x\in \cap_n D_n$, $x\in B \iff x \in U$.

Also well known: every Borel set can be represented by a Borel code, a well-founded tree whose leaves encode open sets and whose inner nodes designate how to combine them (intersect, union, complement).

Taking a computability-theoretic view, it looks like starting from a Borel code $X$ of height $\alpha$, it will take about $\alpha$-many jumps of $X$ to compute open codes for such sets $U$ and $\{D_n\}_{n\in\omega}$.

Do you know of a reference where this (or the analogous fact, if I missed any detail) is already proved?

Similarly, is there any paper which has already analyzed the reverse math strength of the statement ``Every Borel set has the Property of Baire''?

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I think this is a well known fact.

For example, let $\alpha$ be a recursive ordinal and $B=\{g\mid g\mbox{ is a }0^{(\alpha)}\mbox{-generic real}\}$. Then $B$ is a hyperarithmetic set and so has a recursive Borel code $X$.

But no real in $B$ is recursive in $0^{(\alpha)}$ and so any open approximation must not be recursive in $0^{(\alpha)}$.

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  • $\begingroup$ Liang Yu, thank you for the nice example. It is short and pleasing, but different from what I am looking for in two ways. First, the fact I wish to cite would give the exact degree of unsolvability for each $\alpha$, while this example leaves a few jumps' gap. (The gap can be closed by coding $0^{(\alpha)}$ into the $U$.) Second, I'm looking for a reference, not a proof sketch. When I need to cite folklore, I will generally write up a proof for the sake of the literature's completeness. However, I prefer to hope that someone wrote it already! $\endgroup$
    – Linda Brown Westrick
    Commented Apr 5, 2015 at 0:35
  • $\begingroup$ Peter Hinman (Hinman, Peter G. Some applications of forcing to hierarchy problems in arithmetic. Z. Math. Logik Grundlagen Math. 15 1969 341–352. ) investigated something related your question based on Sacks work. $\endgroup$
    – 喻 良
    Commented Apr 5, 2015 at 7:40
  • $\begingroup$ Thank you for the reference. I couldn't find an electronic copy in the usual places, so my response will take a little longer. (If anyone has a pdf handy I would be grateful for the share.) $\endgroup$
    – Linda Brown Westrick
    Commented Apr 5, 2015 at 19:01
  • $\begingroup$ I'm glad to know of this nice paper of Hinman. As for my question, I'll accept your answer of "folklore" in a few days if no references come up before then. $\endgroup$
    – Linda Brown Westrick
    Commented Apr 6, 2015 at 2:02

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