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Is the following (second-order) formula schema provable in ATR$_0$?

Let $\varphi$ be an arithmetical formula satisfying

  1. For all $x, y\in \mathbb{R}$, we have that $x=_\mathbb{R}y$ implies $\varphi(x)\leftrightarrow \varphi(y)$.
  2. For all $x\in \mathbb{R}$, if $\varphi(x)$, then $(\exists N\in \mathbb{N})(\forall y\in B(x, \frac{1}{2^N}))\varphi(y)$.

Then there are sequences of reals $(a_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in >\mathbb{N}}$ such that for all $x\in \mathbb{R}$, we have $\varphi(x)\leftrightarrow (\exists n\in \mathbb{N})(a_n<x<b_n)$.

The idea behind the previous schema is that $\varphi(x)$ defines an open set of reals via 1) and 2), and therefore has a code in the sense of second-order reverse mathematics.

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  • $\begingroup$ Can we assume without loss of generality that $a_n$ and $b_n$ are rational? (I am out of my field of expertise here.) $\endgroup$ Commented Nov 24 at 22:40
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    $\begingroup$ In the context of Type 1 computability, there is a theorem of Moschovakis which proves almost exactly this. See Theorem 11 in Recursive metric spaces, 1964. $\endgroup$ Commented Nov 25 at 13:10
  • $\begingroup$ @AndrejBauer Yes, that is not a problem here. By contrast, the unique representation consisting of pairwise disjoint intervals of an open set does not always have rational end-points. $\endgroup$ Commented Nov 25 at 16:13
  • $\begingroup$ @E.Rauzy Thanks for the interesting suggestion. What Yiannis is studying in that paper seems close to what Dag Normann and I have called "R2-open" sets: in addition to the set $O$ being open, one is also given a function $Y$ such that if $x\in O$, then $Y(x)>0$ and $B(x, Y(x))\subset O$. $\endgroup$ Commented Nov 25 at 19:28
  • $\begingroup$ An uninformed question: is there a relation between above schema and the Lindelöf property? $\endgroup$ Commented 15 hours ago

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