*Fix some computable bijection $b$ between $\omega$ and $2^{<\omega}$. For $r\in 2^\omega$, let $$[r]=\{f\in 2^\omega: \forall\sigma\prec f(b^{-1}(\sigma)\in r)\}$$ be the closed subset of Cantor space coded by $r$. For $M\models PA$ nonstandard, let $\mathcal{S}(M)$ be the standard system of $M$ thought of as a topological space (namely, as a subspace of Cantor space).*

Say that a **closed set pattern** on a topological space $(X,\tau)$ is an assignment $c$ of $\tau$-closed sets to points in $X$.

*EDIT: while it doesn't impact this question or the followup question, it seems natural in retrospect to also add the condition that the relation "$x\in c(y)$" be closed in the product topology; that is, the pattern itself should also be closed.*

Every nonstandard $M\models PA$ has a corresponding closed set pattern on $\mathcal{S}(M)$ given by $$c_M: r\mapsto [r]\cap SS(M).$$ If $M$ is countable the space $\mathcal{S}(M)$ is homeomorphic to the rationals, so any interesting behavior is concentrated on $c_M$.

I would like to understand how closed set patterns of the form $c_M$ behave, and the following seems a good starting point. Say that closed set patterns $c_1,c_2$ on $\mathcal{X},\mathcal{Y}$ respectively are **equivalent** (and write $c_1\sim c_2$) if they differ by a homeomorphism - that is, if there is an $H:\mathcal{X}\cong\mathcal{Y}$ satisfying $$x\in c_1(y)\leftrightarrow H(x)\in c_2(H(y))$$ for all $x,y\in\mathcal{X}$. My question is:

Are there countable nonstandard $M,N\models PA$ such that $c_M\not\sim c_N$?

The knee-jerk approach to a positive answer would be a back-and-forth argument, but since the assignment of closed sets to reals isn't continuous in any good sense that doesn't seem to work here. On the other hand, I don't even see how to start approaching a negative answer.

3more comments