# Proof (or reference) about the cc-ness of termspace forcing

Recall that for $$P$$ a forcing order and $$\dot{Q}$$ a $$P$$-name for a forcing order, the termspace forcing $$T(P,\dot{Q})$$ consists of minimal-rank names for elements of $$\dot{Q}$$, ordered by $$\dot{q}'\leq\dot{q}$$ iff $$1_{P}\Vdash\dot{q}'\leq_{\dot{Q}}\dot{q}$$. In his chapter in the Handbook of Set Theory, James Cummings states the following result:

If $$\kappa$$ is inaccessible, $$P$$ is $$\kappa$$-cc. and $$\dot{Q}$$ is forced to be $$\kappa$$-cc., $$T(P,\dot{Q})$$ is $$\kappa$$-cc.

and attributes it to the paper "More saturated ideals" by Matthew Foreman. However, I was unable to find the above statement in the paper. Later on, Cummings proves it, but only for measurable (or at least Jonsson) $$\kappa$$.

Is there a direct proof (or a different source) for the exact result above?

### Edit:

Philipp Lücke in the comments gave an argument that the stated result is actually false. If $$|P|<\kappa$$ and $$\kappa$$ is weakly compact, it holds. So another interesting question would be:

Does the above statement hold for (not necessarily weakly compact) $$\kappa$$ if $$|P|<\kappa$$?

• The best thing I am aware of is that it holds if $\kappa$ is weakly compact, and $\mathbb P$ is of size less than $\kappa$. Maybe it is true for a poset with the $\kappa$-chain condition. But just with inaccessibility...! Feb 25 at 14:21
• Let $\mathbb{P}$ be the forcing that adds $\kappa$-many Cohen reals, let $\dot{\mathbb{Q}}$ be the $\mathbb{P}$-name for the forcing that adds a single Cohen real and for each $\alpha<\kappa$, let $\dot{q}_\alpha$ denote the canonical $\mathbb{P}$-name for a Cohen condition that is equal to the first digit of the $\alpha$-th Cohen real added by $\mathbb{P}$. Doesn't $\langle\dot{q}_\alpha\vert\alpha<\kappa\rangle$ enumerate an antichain in $T(\mathbb{P},\dot{\mathbb{Q}})$? Feb 25 at 21:16
• It certainly seems to. Thanks. I additionally misspoke in the question: He only proves it in the case when $P<\kappa$. What do you suggest i do with the question? Would you mind adding your comment as an answer? Feb 26 at 7:45
• @HannesJakob You did the right thing! :-) Feb 28 at 9:25
• It seems like it's going to be very difficult to get below a weakly compact, even consistently: Let $\mathbb{Q}$ be a $\kappa$-c.c. forcing notion such that $\mathbb{Q}^2$ is not $\kappa$-c.c. Take $\mathbb P$ be the atomic forcing with 2 atoms, $a_0, a_1$. Let $\{(q_i, q_i') | i < \kappa\}$ be an antichain in $\mathbb{Q}^2$. Take the sequence of terms $\dot t_i$, such that $\dot{t}_i = q_i$ is $\min G_P = a_0$ and $\dot{t}_i = q_i'$ otherwise. This is an antichain in the termspace forcing. Feb 28 at 9:42

Let us look at the property: for every $$|\mathbb P| < \kappa$$, if $$\Vdash_{\mathbb{P}} \dot{\mathbb{Q}}$$ is $$\kappa$$-c.c. then $$T(\mathbb P, \dot{\mathbb Q})$$ is $$\kappa$$-c.c.
It is known that for $$\kappa$$ which is weakly compact, this property holds.
Indeed, if $$A = \langle \dot{q}_i \mid i < \kappa\rangle$$ is an antichain, then for every $$i < j$$ there is $$p\in \mathbb{P}$$ such that $$p \Vdash \dot{q}_i \perp \dot{q}_j$$. This gives us a coloring of pairs of ordinals below $$\kappa$$ with $$|\mathbb P|$$ many color. As $$\kappa$$ is assumed to be weakly compact, there is a homogeneous set $$H$$ with a fixed color $$p$$. So $$p$$ forces $$\langle \dot{q}_i \mid i \in H\rangle$$ to be an antchain in $$\mathbb{Q}$$, contradicting the chain condition hypothesis.
Let us assume that this property holds. In particular, it means that the product of less than $$\kappa$$ many copies of a $$\kappa$$-c.c. forcing $$\mathbb{Q}$$ is $$\kappa$$-c.c.: Let $$\mathbb{P}$$ be the atomic forcing with $$\theta$$ many atoms, and let $$\mathbb{Q}$$ be a $$\kappa$$-c.c. forcing. Then, for every $$\vec q \in \mathbb{Q}^\theta$$ we can assign a name $$\dot{u}$$ which is forced to be $$\vec{q}(\alpha)$$ if and only if $$\min G_P$$ is the $$\alpha$$-th atom. Clearly, this gives us a way to translate an antichain in $$\mathbb{Q}^\theta$$ into an antichain in $$T(\mathbb{P}, \check{\mathbb{Q}})$$.
In the paper, "Knaster and friends I: Closed colorings and precalibers", by Lambie-Hanson and Rinot, they define the combinatorial principle $$U(\kappa,\mu,\theta,\chi)$$. Let us focus on the case $$U(\kappa, 2, \omega, 2)$$. They proved that this case (and stronger ones) holds for successor cardinals, or if $$\square(\kappa)$$ holds [so in $$L$$, it fails exactly for weakly compact cardinals]. Moreover, they show (for example) that $$U(\kappa,2,\omega,2)$$ implies the existence of a $$\kappa$$-c.c. forcing which its $$\omega$$-th power is not $$\kappa$$-c.c., and they conjecture that $$\neg U(\kappa,2,\omega,2)$$ implies that $$\kappa$$ is weakly compact.