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?


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 $|P|<\kappa$?

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    $\begingroup$ 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...! $\endgroup$
    – Rahman. M
    Feb 25 at 14:21
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    $\begingroup$ 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 $T(\mathbb{P},\dot{\mathbb{Q}})$? $\endgroup$ Feb 25 at 21:16
  • $\begingroup$ 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? $\endgroup$ Feb 26 at 7:45
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    $\begingroup$ @HannesJakob You did the right thing! :-) $\endgroup$
    – Rahman. M
    Feb 28 at 9:25
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    $\begingroup$ 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. $\endgroup$
    – Yair Hayut
    Feb 28 at 9:42

1 Answer 1


The answer is consistently negative, and it seems likely that it is actually always negative.

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.


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