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.

  • $\begingroup$ 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 $RT_2^2$. $\endgroup$ – James Hanson Feb 29 at 17:24
  • $\begingroup$ @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. $\endgroup$ – 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 $$\{\sigma\in 2^{<\omega}: \vert\sigma\vert<i\mbox{ or }\sigma\prec f\}.$$ 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 $2^{<\omega}$.

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  • $\begingroup$ 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. $\endgroup$ – Noah Schweber Feb 29 at 17:54

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