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For $\mathbb{A},\mathbb{B}\subseteq\mathcal{P}(\omega^\omega)$, say $\mathbb{A}$ spreads onto $\mathbb{B}$ iff there is some $F:\omega^\omega\rightarrow\omega^\omega$ such that for all $X\in\mathbb{A}$ the set $F[X]=\{F(r):r\in X\}$ is in $\mathbb{B}$.

Let $\mathbb{D},\mathbb{E}$ be the sets of dominating, escaping families of reals respectively. Trivially $\mathbb{D}$ spreads onto $\mathbb{E}$ via the identity function, since every dominating family is escaping. The converse however is more complicated, even in $\mathsf{ZFC}$: $\mathbb{E}$ obviously can't spread onto $\mathbb{D}$ if $\mathfrak{b}<\mathfrak{d}$, but $\mathsf{CH}$ implies that $\mathbb{E}$ does spread onto $\mathbb{D}$ per vzoltan's answer to an old question of mine.

I'm curious what happens in $\mathsf{ZF+DC+AD}$:

Under $\mathsf{ZF+DC+AD}$, does $\mathbb{E}$ spread onto $\mathbb{D}$?

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  • $\begingroup$ If escaping means unbounded, then the answer should be no, since you can force $\mathfrak{b} < \mathfrak{d}$ over a model of ZF + DC + AD without adding reals. It should be possible to make this work just from the fact that if every set of reals has the property of Baire, then every function from the reals to the reals is continuous on a comeager set. $\endgroup$ Jun 5 at 20:47
  • $\begingroup$ @PaulLarson Yes, escaping means unbounded. How do you force $\mathfrak{b}<\mathfrak{d}$ over a determinacy model without adding reals? (Maybe the answer is that I should read your book. :P) $\endgroup$ Jun 5 at 20:48
  • $\begingroup$ With a $\mathbb{P}_{\mathrm{max}}$ variation. But there has to be a simpler way. $\endgroup$ Jun 5 at 20:52
  • $\begingroup$ @PaulLarson Oh huh, that's not what I expected. Where would I find the details for that? (In your stationary tower book?) $\endgroup$ Jun 5 at 20:54
  • $\begingroup$ The Shelah-Zapletal paper "Canonical models for $\aleph_{1}$-combinatorics" is probably the best place. Anyway, I will try to write a $\mathbb{P}_{\mathrm{max}}$-less argument soon. $\endgroup$ Jun 5 at 21:03
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Here's something that seems to work. I can add more details if needed.

Suppose that every set of reals has the property of Baire. Then every function from $\omega^{\omega} \to \omega^{\omega}$ is continuous on a comeager set. Note that every comeager set is unbounded. Fix $F \colon \omega^{\omega} \to \omega^{\omega}$ and let $C$ be a comeager set on which $F$ is continuous. We will find a dense open sets $D_{i}$ ($i \in \omega$) and $y \in \omega^{\omega}$ such that $F(x)$ fails to dominate $y$ whenever $x \in C \cap \bigcap_{i \in \omega}D_{i}$. Let $\langle (i_{n}, \sigma_{n}) : n \in \omega \rangle$ be an enumeration of $\omega \times \omega^{<\omega}$. In the $n$th stage of the construction choose $y \upharpoonright k_{n}$ for some $k_{n} \in \omega$ and also some $\tau_{n} \in \omega^{<\omega}$ extending $\sigma_{n}$ such that $F(x) \upharpoonright k_{n}$ is the same sequence $\rho_{n}$ for all $x \in C$ extending $\tau_{n}$, and $\{ j < k_{n} : \rho_{n}(j) < y(j)\}$ has size at least $i_n$. Then we can let $D_{i}$ be the set of $x \in \omega^{\omega}$ extending any of the sets $\tau_{n}$ where $i_{n} = i$.

The $\mathbb{P}_{\mathrm{max}}$ claim has to be modified somewhat, since $\mathbb{P}_{\mathrm{max}}$ needs AD$^{+}$ to work, which is not known to follow from AD + DC. Forcing over a model of AD$^{+}$, $\mathbb{P}_{\mathrm{max}}$ produces a model of ZFC which is $\Pi_{2}$ maximal for the powerset of $\omega_{1}$ by (very roughly speaking) forming a generic direct limit of all countable models, subject to agreement about stationarity for subsets of $\omega_{1}$. There is a natural $\mathbb{P}_{\mathrm{max}}$ variation for each $\Sigma_{2}$ sentence, where each model fixes a witness to the sentence, and a model in the direct limit passes its witness on to stronger conditions. This doesn't always succeed, but for the statement $\mathfrak{b} = \aleph_{1}$ it does succeed, producing (without adding reals) a model of $\mathfrak{b} = \aleph_{1}$ in which every $\Pi_{2}$ sentence for $\mathcal{P}(\omega_{1})$ holds which can be provably forced to hold along with $\mathfrak{b} = \aleph_{1}$. For instance, the statement $\mathfrak{d} > \aleph_{1}$. This is discussed at a general level in the last section of my article from the Handbook of Set Theory, and in more detail in the Shelah-Zapletal paper "Canonical models for $\aleph_{1}$-combinatorics".

Having produced such a model, one can then take an unbounded family $B$ of size $\aleph_{1}$, find a real $y$ which is not dominated by $F[B]$, and then find the unbounded set of $x$ such that $F(x)$ does not dominate $y$ back in the ground model.

There is an intermediate argument one could run using the fact that, under AD$^{+}$, true $\Sigma^{2}_{1}$ sentences have Suslin, co-Suslin witnesses. This is at the heart of the $\mathbb{P}_{\mathrm{max}}$ argument. For the current problem, continuity on a comeager set works as a substitute for being Suslin and co-Suslin.

Morally, anything that is consistent with large cardinals should be forceable over a determinacy model without adding reals.

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  • $\begingroup$ This is really helpful, thanks! (I've taken the liberty of adding links to the papers you mention.) $\endgroup$ Jun 6 at 19:12

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