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Below work in $\mathsf{ZFC+CH}$ for simplicity.

Say that a (set) forcing notion $\mathbb{P}$ captures a map $f:\mathbb{R}\rightarrow\mathbb{R}$ iff there is some $\mathbb{P}$-name for a real $\nu$ such that the following is forced: "For all $r\in V$ we have $f(r)\le_T\nu\oplus r$." (Here "$f(r)$" is computed in $V$.) Every function is captured by $Col(\omega,\omega_1)$ since we can take $\nu$ to simply be an array coding the graph of $f$, so I'm interested in capturing functions by forcings which don't collapse $\omega_1$. For example, the jump and hyperjump are so captured: Hechler forcing $\mathbb{H}$ captures the jump (and double-jump and etc.), and its iterate $\mathbb{H}*\dot{\mathbb{H}}$ captures the hyperjump. Jump-capturing is just via the relativized Busy Beaver function, while hyperjump-capturing uses a neat trick: if $h$ is Hechler-generic then $(r\oplus h)''\ge_T\mathcal{O}^r$ since an $r$-computable tree $T\subseteq\omega^\omega$ is well-founded iff for all finite modifications $\tilde{h}$ of $h$ the part of $T$ to the left of $\tilde{h}$ is well-founded (and this part of the tree is explicitly finitely branching so well-foundedness can be determined via a single jump).

My question is whether this is in fact trivial:

Is it consistent with $\mathsf{ZFC}$ that there is a function which is not captured by any $\omega_1$-preserving forcing?

(I would also be happy to replace "$\omega_1$-preserving" with "proper" or similar, if that would make things more interesting.) A natural (family of) candidates is something like $f(r)=$ the $L$-least real coding a model of $T$ with $r$ as an element, under the assumption $\mathsf{V=L}$, but I don't actually see how to show this can't be so captured.

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    $\begingroup$ If you iterate Hechler $\omega_1$ times, what then? $\endgroup$ Commented Dec 1 at 7:00
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    $\begingroup$ What's the single (name for a) real $\nu$ that you're looking at? (Maybe it's more natural to iterate $\omega_1+1$ times, and then look at the final Hechler real?) At a glance, though, you just keep climbing the iterated-hyperjump hierarchy; I don't see that anything like that even gives you the 2-hyperjump (i.e. the map sending a real $r$ to the canonical complete $\Pi^1_2(r)$ set). The class of $\Delta^1_2$ operators is huge. $\endgroup$ Commented Dec 1 at 7:42
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    $\begingroup$ Hm. What if you iterate Hechler reals $\omega_2$ times? What if you use almost joint coding to code functions into reals and then just iterate through the ground model functions? $\endgroup$ Commented Dec 1 at 10:45
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    $\begingroup$ Out of curiosity, where can I find the trick about the Hechler real and an $r$-computable tree you mentioned? $\endgroup$ Commented Dec 1 at 17:59
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    $\begingroup$ @AsafKaragila I've deleted my previous comment, I think I just don't know enough about AD-coding to follow this at the moment. I'll definitely look at it later today though (this has been on my list of things to learn for a sadly long time). $\endgroup$ Commented Dec 1 at 20:21

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No, for any function $f$ there is a c.c.c. forcing $\mathbb P_f$ so that $f$ is captured by a real in $V^{\mathbb P_f}$. I will use $\omega^\omega$ instead of $\mathbb R$.

The standard construction of an almost disjoint family $\{a_r\mid r\in\omega^\omega\}\subseteq \mathcal P(\omega)$ of size continuum has the nice property that $a_r\leq_T r$. There is now a c.c.c. forcing $\mathbb A_f$ which adds $x\subseteq\omega$ so that for any $r\in V$, $$f(r)(n)=m\Leftrightarrow a_{n^\frown m^\frown r}\cap x\text{ is infinite}.$$ $\mathbb A_f$ is simply a suitable instance of almost disjoint coding forcing. Hence $f(r)\leq_T (x\oplus r)''$ for any real $r\in V$.

We can now use your Hechler forcing trick: if $y$ is Hechler-generic over $V[x]$ then for all $r\in V$ we have $(x\oplus r)''\leq_T y\oplus x\oplus r$. So $\nu=y\oplus x$ works in $V^{\mathbb P_f}$ for $\mathbb P_f=\mathbb A_f\ast\dot{\mathbb H}$.

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    $\begingroup$ Neat, so my intuition was right. $\endgroup$ Commented Dec 2 at 17:06
  • $\begingroup$ Sorry for not contributing with anything but where can I read about $\mathbb{A}_f$ kinds of forcing? $\endgroup$ Commented Dec 3 at 19:23
  • $\begingroup$ @AsafKaragila Yes, indeed! $\endgroup$ Commented Dec 3 at 20:23
  • $\begingroup$ @H.CManu Almost disjoint coding forcing was first used by Solovay and first published in the Martin's Axiom paper "Internal Cohen extensions" by Martin-Solovay. It is used to show that if $\mathrm{MA}$ holds and $\kappa<2^\omega$ then $2^\kappa=2^\omega$. This is Theorem 16.20 in Jech's Set Theory. It is also a crucial ingredient in Jensen's Coding the Universe theorem. $\endgroup$ Commented Dec 3 at 20:29
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    $\begingroup$ Btw I just found a subsection on this type of forcing in the Chong and Yu book(Higher Recursion) and they attribute it to Silver, why do you guys think that is the case? $\endgroup$ Commented Dec 4 at 7:59

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