# Detecting comprehension topologically

This question basically follows this earlier question of mine but shifting from standard systems of nonstandard models of $$PA$$ to $$\omega$$-models of $$RCA_0$$. For $$X$$ a Turing ideal we get the map $$c_X$$ on $$2^\omega$$ given by $$c_X(x)=[b[x]]\cap X$$ where $$b$$ is some computable bijection $$\omega\cong 2^{<\omega}$$; in the language of the linked question, this is a closed set pattern.

I'm trying to understand how much information this construction loses in the sense of second-order arithmetic. For $$X$$ a Turing ideal we also get an $$\omega$$-model of $$RCA_0$$ - which I'll freely conflate with $$X$$ itself - and I'm interested in what axioms of second-order arithmetic we can detect topologically.

(Below, $$\sim$$ denotes "difference by homeomorphism" - for $$c,d$$ closed set patterns on $$\mathcal{X},\mathcal{Y}$$ we write $$c\sim d$$ iff for some $$H:\mathcal{X}\cong\mathcal{Y}$$ we have $$a\in c(b)\leftrightarrow H(a)\in d(H(b))$$ for all $$a,b\in\mathcal{X}$$.)

Say that a theory of second-order arithmetic is topologically detectable if there is some $$\sim$$-respecting property of closed set patterns which holds of $$c_X$$ iff $$X\models A$$ for each Turing ideal $$X$$. For example:

• $$WKL_0$$ is topologically detectable via "$$c_X^{-1}(\emptyset)$$ is open."

• $$ACA_0$$ is topologically detectable in at least two ways (after adding $$WKL_0$$): via "$$ran(c_X)$$ is closed under (single) Cantor-Bendixson derivatives" and "for each $$x\in X$$ the set $$\{y\in X: c(y)\supseteq c(x)\}$$ is closed."

My question is essentially whether any other "weak combinatorial principles" are topologically detectable:

Is there some "reasonably natural" $$A$$ with $$ACA_0\models A$$ and $$WKL_0\not\models_\omega A$$ such that $$A$$ (or at least $$WKL_0+A$$) is topologically detectable?

(Here $$\models_\omega$$ is the restriction of $$\models$$ to $$\omega$$-models, which is needed to rule out e.g. $$I\Sigma_{17}$$. Note that if $$A$$ is topologically detectable then so is $$WKL_0+A$$, but the converse isn't obvious to me.)

The most tempting candidate is of course $$RT^2_2$$, but there are plenty of others.

• Is there a good list of things that are know to follow from $ACA_0$ but not $WKL_0$? All of the examples I can find seem to be related to $RT_2^2$. – James Hanson Feb 29 at 17:24
• @JamesHanson Re: your first question, at a glance they don't need that but I was being lazy - we can always throw on "... and $\{a: c_X(a)=\emptyset\}$ is open" to whatever characterization we have, as long as we're shooting for a principle above $WKL_0$. Re: your second question, yes the paradigm is Ramsey theory - the reverse math zoo is a good source. – Noah Schweber Feb 29 at 17:24

Statements about existence of $$\omega$$-models can be topologically detected.

Specifically, fix $$X$$ a Turing ideal. For $$t\in X$$ say that $$t$$ enumerates a family of sets if:

• Exactly one $$p\in c_X(t)\cap X$$ has $$c_X(p)=X$$.

• For every other $$q\in c_X(t)$$ we have $$c_X(q)=\{a\}$$ for some $$a\in y$$.

• For each $$a\in y$$ there is exactly one $$q\in c_X(t)$$ with $$c_X(q)=\{a\}$$.

In such a case we say $$t$$ enumerates the family $$X_t:=\{q\in X: \exists a\in c_X(t)(c_X(a)=\{q\})\},$$ and we can talk about the induced closed set pattern coming from $$X_t$$. It's not hard to see$$^*$$ that every sequence of sets in $$X$$ (that is, the whole sequence is in $$X$$) corresponds to such an $$X_t$$, and this means:

If $$A$$ is a topologically detectable sentence, so is the statement $$O_A$$ = "Every real is contained in an $$\omega$$-model of $$A$$."

Taking $$A=WKL_0$$ then gives an affirmative answer to the question. Of course $$RCA_0+O_{WKL_0}\vdash WKL_0$$: if $$X$$ is an $$\omega$$-model of $$RCA_0$$ and $$T$$ is an infinite binary tree in $$X$$, then any $$\omega$$-model of $$WKL_0$$ containing $$T$$ also thinks $$T$$ is an infinite binary tree - and being a path through a tree is absolute between $$\omega$$-models. (More generally, we have $$RCA_0+ O_\varphi\vdash\varphi$$ for every $$\varphi\in\Pi^1_2$$.)

Meanwhile, $$RCA_0+O_{RCA_0}\vdash WKL_0$$, since from a coded $$\omega$$-model of $$RCA_0$$ we can whip up a $$DNR_2$$ function (and this relativizes). So this approach does not produce an example of a topologically detectable sentence incomparable with $$WKL_0$$.

$$^*$$Specifically, given a sequence of reals $$F=(f_i)_{i\in\omega}$$ let $$s_i$$ be the natural code for the tree $$\{\sigma\in 2^{<\omega}: \vert\sigma\vert The closure of $$\{s_i: i\in\omega\}$$ is the set of paths through a pruned tree $$T$$ - which is computable in $$F$$ - and $$[T]$$ has only one "extra" path, corresponding to $$2^{<\omega}$$.

• Of course this isn't totally satisfying - something of Ramseyan flavor would be better, and I'll hold off on accepting this answer in the hopes of a better answer - but it does give something. – Noah Schweber Feb 29 at 17:54